Just to give you an idea of the extraordinary reactions this piece can provoke, here is a website entirely devoted to the piece and its influence. That is just for interest, I don't advocate any of the material or claims on that website, which get rather too ephemeral. But it would be interesting to have a look at the music. It's only two pages, so I'll put up the whole thing:
The time signature nowadays would be written 2/2 or the half-circle with a slash that is often called "cut time" but that is really an archaic medieval modal time signature meaning tempus imperfectus or duple time. The key is B flat. It sounds almost as if it starts in the middle, which is a nice effect. Composers can go one of two ways to avoid the opening of a piece sounding too standard: they can compose an introduction, often in a slower tempo, or they can, as novelists sometimes do, give the effect of coming in in the middle of the story, which Couperin chooses.
The harmonic effect here is one that comes from counterpoint, often called "voice-leading". You might remember from my post of a few days ago I cited a book that I called a "bible of music theory", Aldwell and Schachter's Harmony and Voice-Leading? Well, it is pieces like the Couperin that are responsible for why the phrase "voice-leading" is in the title. Harmony actually, both historically and currently, comes from counterpoint, from voice-leading. The phrase "voice-leading", by the way, just refers to the way the voices move from harmony to harmony (from one chord to the next). But harmony itself is built from different voices. One of the problems with harmony nowadays is that a lot of composers and listeners have forgotten this. In the strummed chord progressions of popular music, the contrapuntal nature of harmony is nearly extinguished.
But the whole effect of Couperin's Les Baricades Mistérieuses depends on the voice-leading. The kind of texture he is creating uses two different kinds of effects. In one, a voice is held over from a previous harmony and resolves down into the next harmony after the bass note is sounded. There is an example in the very beginning where the top voice sounds a D, the third of the tonic B flat, and this is held over the sounding of the F in the bass, the root of the dominant F chord. It then resolves down by step to the C, the fifth of the dominant. In modern harmony courses, this is called a suspension, specifically a 6-5 because those are the intervals over the new bass note. In French baroque music this is often indicated with an ornamental sign and is called a coulé. Also at the beginning, but in the alto voice, is another suspension, from B flat to A, a 4-3 suspension in modern terms. Then, going into the next measure, after the repeat sign, the A now resolves up into the B flat when the harmony returns to the tonic. For some odd reason, modern harmony books, even Aldwell and Schachter, focus almost entirely on the suspension that resolves down. But Couperin makes considerable use of the reverse: a dissonance that resolves up, usually the leading tone resolving to the tonic, what we might call a 7-8 resolution. Theorists seem to want to regard this as an anomaly, but Baroque composers seem quite comfortable with it. The term for this kind of ornament is port de voix or "carry the voice". In the next measure we see a 9-8 suspension, a 4-3, and going into the next measure again, a 7-6.
The lovely and mysterious effect that Couperin creates here is through the inventive and layered use of suspensions of both kinds, the standard one that resolves down and the less-common one that resolves up. I say "layered" because he often has two different suspensions resolving at different times in the same measure. It creates a kind of filigree of harmony. All this depends on a contrapuntal device however! This piece is not a simple succession of chords, it is a harmonic web of different voices that resolve according to long-standing rules of counterpoint. The most important: dissonances resolve by step, whether up or down. Probably the reason for this is that when the note resolves by step, we know exactly where it came from, i.e. which voice it is. Bach sometimes breaks this rule by resolving a leading tone down an octave, for example, but he always does it in a context where it is clear what note is being resolved where.
One final comment, another kind of layer here is that while one voice is resolving by being delayed by an eighth-note, another might be resolving after a quarter-note delay. There are different levels of layers!
Let's listen to that again, this time trying to hear all the suspensions. Here is a version on piano played by Cziffra György: