“…it awakens feelings that are not far from those caused by euphony.”

From this article, which I’ll cite also below

This work is perhaps the most traditional of the quartets we’ll be discussing, and leave it to Schoenberg to produce a work that at once embodies such traditional and modern qualities.
Perhaps not actually his last quartet to be completed, though, it is number four. The Wikipedia article that sweepingly addresses all four of the quartets mentions in unsatisfying passing that there were many movements, attempts, or incomplete things written for string quartet, and mentions a string quartet no. 5 from 1949. A Google search produced this link from Belmont Music Publishers with a video that I can’t access, perhaps due to my location, or the video is just gone.
In any case, no. 4 from 1936 is the last published, ‘official’ quartet from Schoenberg, from around the same time as his violin concerto, and if you’re familiar with that work or his others of the time, this quartet fits pretty clearly into his late style.
The piece was a commission from (for?) Elizabeth Sprague Coolidge, who’d also commissioned his third quartet, but the fourth came at a rough time in Schoenberg’s life. A busy schedule and the weather affected his health, and thereby the family’s financial situation; they moved during this time from New York to California, not to mention the whole “leaving your home country because of World War II” situation. But apparently the work itself was completed in a rather short period of time.
The piece is in four movements, like you’d expect from Beethoven or Brahms or anyone else, even with a sonata-form type layout for the first movement. The second movement kind of sort of lilts and tips at a waltz, followed by a slow, dark third movement and what many people, including the Brentano Quartet’s program notes for the piece describe as a march-like finale. We’ll get to that.
It’s important that you’ve read the articles about some of Schoenberg’s other works: Verklärte Nacht, Gurre-Lieder, his Drei Klavierstucke; is there more? In any case, as the history books have told us and sufficiently boiled down to a nice clean statement, Schoenberg was the man who ‘invented’ twelve-tone composition (I know, I know), but he was also a die-hard Romantic. So what I personally find interesting in this quartet, as we shall discuss, is that, like Webern, he here uses a very traditional, classical form (the string quartet, in four movements) to present
to us a solidly, fully-formed, mature twelve-tone work, but one that is kind of elegantly and seemingly effortlessly drawn up into a clean, Classical package. How cool is that?
One of the big lightbulb questions I had when I began to think about and process twelve-tone music was “How do you present a sonata-form structure outside of the context of diatonic harmony?” Without the minor/major or tonic/dominant relationship to the two main themes of a sonata form, how does one present that structure so that there’s a ‘plot’ to a movement?
Had I heard this piece at the time (at least a year ago by now), I would have probably been able to identify the two themes at play, but possibly not why they were identifiable. In any case, there are many different ways a composer can lay out the two main subjects of his work: certain intervals within the row, certain rhythms or executions of the same row, and as we saw with Babbitt last week, even something like the dynamics for certain rows becomes a factor in making up the personality of a subject.
What Schoenberg starts with, again in contrast with Babbitt’s penchant last week for hiding and kind of obscuring the row, is the row pretty much laid out for us in plain sight. This maybe is a different kind of suspense, like when you’re watching a thriller action film and you also don’t know who the bad guy is; that’s one kind of suspense. But even if you have an omniscience that the character doesn’t, you still feel interested and engaged. There’s tension.
I thought there was a jpg of the first few bars of the quartet somewhere online, but I can’t seem to find it, so check out this article called A Primer for Atonal Set Theory, which seems like a good read by itself, but also shows the opening few bars for first violin. (It also delightfully gives credit first to Milton Babbitt for development of the idea while acknowledging that lots of people did similar things in different ways.)
In contrast with Babbitt’s quartet (or his other works), the first twelve pitches that the violin plays make up the row. Note a few of them repeat when they appear, but the order of the pitch classes is not affected. This opening melody, to be honest, I find to be mildly grating, bristly, and not terribly pleasant. Whether that is the intended effect or not is irrelevant; that opening line, and probably some specific intervals or places within it, feel almost immediately seared into the brain, so that when similar melodies or progressions pop up again later (read the atonal set theory article, maybe), they’re quite easily identifiable. And pop up they do, and not just in this movement. In any case, this is a very effective way to begin the piece, quite outright, but at least transparent. The article linked in the opening quote is here, and it describes the the first movement as such:

The basic row is presented in the first five bars; starting in bar six it is contrasted with a lyrical secondary theme. Over the course of the movement, lyrical episodes develop from this theme, which is repeatedly contrasted with the striking main theme. In addition to the basic row, which is notable for its tone repetition, thirds and sixths are heard not only as horizontal intervals but also as chordal elements of the accompaniment.

The twelve notes of the row are broken into four trichords (I saw this somewhere online and can’t seem to find it now), certain tonal qualities built into the row when played simultaneously rather than one-after-the-other as in the opening. Naxos, on this sparse but informative page, I believe is quoting the composer himself as saying:

[The first violin’s opening phrase] can rightfully be called the main theme, because of its frequent recurrences, some of which one might be inclined to consider as recapitulations in the manner of the sonata form.

I think that’s quite noticeable to most ears. In any case, those two above sources explain the piece very well. What’s satisfying about the first movement is that, even for a new listener, it feels like it outlines important things to listen to, has a readily accessible structure and focus, and even if it isn’t whitebread tonality from the Classical era, it should be easy to follow.
After the longest movement, we have the second, marked comodo. Aside from taking the role of the minuet and trio (I guess? It’s in 3/4, and ternary form), the piece feels strongly related to the opening movement, at least at the beginning. It feels like we haven’t traveled very far, but rather made a few changes. We move farther away from the content of the first movement in the B section, but themes from the opening subject do reappear.

The real change, I feel, breathtaking, like a splash of cold water (be that good or bad) is in the third movement, marked largo, and the shortest of the four. It’s rich and powerful, but also seemingly delicate and frail. After a long, slow unison passage, the cello is the first to break the form, and it’s kind of sad and beautiful, a pained slow movement, written in 8/8 with quarter note equalling 78.
Naxos, linked above, quotes the composer as saying:

In six measures a climax is reached by semicontrapuntal elaboration and development of the contents of these two measures, which is dissolved into a segment, bridging and introducing the recapitulation of the B section.

In any case, the B section mentioned here is quite obvious in its motivic nature, lively compared to the opening A section, which, again splashingly, reappears around halfway through the movement. That whole unison thing in octaves is powerful stuff, and it’s used to great effect. Again the cello is first to appear, in a “compressed recapitulation of the B section.”
The final movement is what many refer to as the march, but if anything, I feel it calls to mind the first movement, which felt more inspiringly marchy. But that doesn’t last long, and it becomes quickly apparent that this movement is the most lively and perhaps intense of them all. There are short bursts of energy, almost chaos, in a 6/8 section marked agitato. Interesting things happen in this movement like the violins being marked in 6/4 against the lower strings in 12/8, the same length, just double against triple meter. Schoenberg himself says of this movement: “This Allegro contains a great abundance of thematic material because every repetition is varied far-reachingly and gives birth to new formulations.”
It seems to be the tense climax of this entire, quite troubled yet delicately beautiful quartet. There are moments that sound like something out of a Hitchcock film score (I know, I know),  and then this interesting little bit; surely plenty to be occupied with, but I at least hear some relation to the first movement, making for a tidy, emotional, disturbed package. The movement (and the piece) ends delicately and unassumingly, almost peacefully, if you were inclined to believe it.
In this paper about the piece that I didn’t entirely read, written by Nathan Stolz, the ultimate conclusion it seems, is as follows, from the Conclusion section on page ten:

It seems Schoenberg uses a particular technique for a short phrase, say two to five measures, and then he switches to another technique. He rarely continues a single contrapuntal technique for an extended period of time, and therefore he employs many. Schoenberg uses different partitioning of the row to create new melodies. In the case of this row, it is necessary to do so, because the constant division of the row into the same four trichords can be limiting.