I wanted to learn a good opening against 1.d4 that does not involve the second move being 1...d5. I personally don't like symmetric positions very much. To give you an idea of the type of openings I play right now, for white, I play Catalan and I play French against 1. e4. Anyone have any ideas for responses I should look into?
Try the Grunfeld or King's Indian Defense. 1. d4 Nf6 2. c4 (usually the follow-up, although 2. Nf3 is also common) 2...g6 3. Nc3 is the start of both lines, then either 3...d5 for the Grunfeld or 3...Bg7 for the King's Indian.
Although the Grunfeld involves ...d5, it is by no means symmetric (usually). The main line continues 4. cxd5 Nxd5 5. e4 Nxc3 6. bxc3, after which White has a massive pawn center that Black seeks piece play against. The King's Indian is, of course, not symmetric either.
You could also play the Nimzo-Indian, a very respectable opening. This also begins 1. d4 Nf6 2. c4, but now you play 2...e6, intending to meet 3. Nc3 with 3...Bb4. If White plays 3. Nf3, you can try the Queen's Indian Defense 3...b6.
I suggest c5. As commented by @NoseKnowsAll it is called the Old Benoni Defense. If white is a pro they will probably have an edge because of their experience but generally, for your criteria (it is not symmetrical) it has the highest statistics for black winning in some of the databases such as https://www.365chess.com/opening.php?m=2&n=7&ms=d4&ns=7. Note that it is moderately common (+16k games in the database cited above) which makes the statistics more meaningful than some uncommon moves like a5.
i dont know how strong it is but i always play the budapest gambit 1.d4,Nf6 2.c4,e5!? 3.dxe5,Ng4 4.Nf3,Nc6 5.Bf4,Bb4+ 6.Nbd2,Qe7 for a few different reasons. the best of these reasons is a trap that ends in checkmate. ive won at least 50 games with it. after 7.a3?,Ngxe5 8.Bxe5,Nxe5 9.axb5??,Nd3#.