Yes but there is a difference between unsound and flat-out bad. Unsouond usually means there is a refutation, while bad moves refute themselves. I guess 1...d5 can be called unsound, but it's not likely to give Black anything even if White has no idea what to do.
I much prefer 1.c4 b5 to 1.c4 d5. It may not be sound but at least it doesn't drop central pawns!
OK what about 1...b5 and then after white takes b5 play 2...a6 with an accelerated Benko gambit? 
In the master database black has 2 wins in this accelerated Benko gambit...wow looks dynamic 
Wow. 2 wins? Impressive. I got that many in the two blitz games I played with 1.e4 e5 2.Nf3 f6, so in my database that has 2 wins too. One of them went 3.Nxe5?! (3.Bc4! is much stronger, feel free to argue with me pfren but I'm pretty confident on this one :) ) 3...Qe7 4.Qh5+?? g6 5.Nxg6 Qxe4+ and Black won easily, and the other went 3.d4?! d5! 4.dxe5 dxe4 5.Nd2 e3! 6.fxe3 fxe5 with a roughly equal position which I later one.
A lot can be played against the English. None of it is really sharp if white won't commit to a plan. So this is my input, granting some assumptions.
[Uitslag "*"]
[ECO "A31"]
[Opening "English: symmetrical variation"]
1. c4 c5 2. Nf3 Nf6
[Instead the game Aronian-Nakamura went as follows: 2... Nc6
3. d4 cxd4 4. Nxd4 Nf6 5. g3 e5 6. Nb5 Bb4+ 7. Bd2 a6 8. N5c3 d5 9. cxd5 Nxd5
10. Bg2 Be6 11. O-O
No pawn sacrificed and I can't figure what is wrong with the black position.]
3.
d4
Not forced, but still a pretty natural move to gain a space advantage. [3.
Nc3 If white is not willing to take the initiative, there is the following: 3...
d5 4. cxd5 Nxd5 The Rubinstein. 5. g3 Nc6 6. Bg2 Nc7 7. d3 e5
Sets up a reversed Maroczy Bind as if (s)he were white! 8. O-O Be7 9. Nd2 Bd7
10. Nc4 O-O [Or 10... f6 With an interesting game.]]
3... cxd4 4. Nxd4 e5 5. Nb5
d5 6. cxd5 Bc5 7. N5c3 O-O 8. e3 e4
Black is active, has space, and it remains to be seen if white can really hold on to the d5 pawn.
*
The problem with 1...d5 though is 1.d4!, transitioning into a queen's gambit. I wouldn't recommend unless you know how to play 1.d4,d5 lines as black. Would 1...f5 be good against the English? The only problem I see is that a pawn on d3 could be a problem for black in many lines, since a knight at d2 can blast open the center with lasting initiative with e4, even gambiting the pawn in some cases.
OK what about 1...b5 and then after white takes b5 play 2...a6 with an accelerated Benko gambit?
In the master database black has 2 wins in this accelerated Benko gambit...wow looks dynamic
It's a stupid gambit, as Black has none of the static targets he has in the Benko.
Factly, black is a pawn down for nothing- not even a mild developmental advantage.
Really I think your best bet would be playing a hedgehog-type setup in the Symmetrical English... it's very sound.
I think really you don't need something unsound for it to be good.
So no, don't use something obscure!
c4 Nc6 I don't believe is good... you might get a bad transposition to Chigorin's Defense...
I play the English regularly, and would be more than happy to transpose to a Chigorin
One very interesting and totally sound line is in the symmetrical and offers very interesting play:
Hmmm... I have never seen that idea either.
If you're looking for obscure stuff that's interesting, check out the games of Hikaru Nakamura.
Here's one game he played:
Hmmm... I have never seen that idea either.
That was in reference to the Symmetrical English.
If you find the Symmetrical English to be "obscure" and have never seen it before, perhaps this thread should be renamed?
No, I know the Symmetrical just fine, just not the Qd7 move Schevenadorf posted.
I have played both sides of the Symmetrical English.
Nakamura in that game had development issues. When I saw 18.Bxb8 I almost facepalmed because of trading a great minor piece for an undeveloped one, and then I saw the fork and was like, "that makes sense". Was that a bullet game?
The problem with 1...d5 though is 1.d4!, transitioning into a queen's gambit. I wouldn't recommend unless you know how to play 1.d4,d5 lines as black. Would 1...f5 be good against the English? The only problem I see is that a pawn on d3 could be a problem for black in many lines, since a knight at d2 can blast open the center with lasting initiative with e4, even gambiting the pawn in some cases.
1...f5 is perfectly fine though I prefer 1...e5 first, then 2...d6, then 3...f5
It is, though my mind somehow shifted to 1.Nf3,f5 2.d3 positions. There is a lot of crossover between the English and 1.Nf3 so it's natural. Still, 1...f5 is playable. 1...e5 looks good too as it creates a hypermodern vs. classical game with contrasting strategies and therefore imbalances.
Schiller is a terrible author and his UCO book is probably his worst, which is saying an awful lot.
These gambits are much worse than the related ones from the Scandinavian. Imagine playing the Smith-Morra down a full tempo (1.e4). Why would Black have reasonable compensation after 1.c4 d5 2.cxd5 e6 3.dxe6 Bxe6? White can continue by putting his pawns in the center after which Black has nothing and White has no targets to attack comfortably. Black has given up both central pawns for no good reason.
I agree with all of that. Well, Schiller isn't a terrible author but anyway...
But the OP asked for something UNSOUND