A logical question is what black can do in this version compared to the regular centre attack.

1.e4 e5 2.Nf3 Nc6 3.Bb5 Nf6 4.d4 Nxd4 5.Nxd4 

and now, instead of recapturing the knight, try and exploit the non-standard position of the bishop with 5...c6. After 6.Bc4 exd4, the threat of Nxe4 is much stronger compared to the regular centre attack because the knight can be reinforced with ...d5.

I don't claim that black is any better here. I'm just trying to figure out ways in which black can avoid the centre attack themes.

Interesting line, thanks for sharing!

Saw a thread on this move years ago and thought I'd add my new 2 cents, though it's probably not theoretically relevant to the GMs and correspondence players of the world. Another big thank you to the forum--I've read so much and contributed so little, so here's something:

If 4...exd, continue in center attack manner. For me I like to castle and play Re1 before e5 so the knight can't hop to e4 then c5. Sure the "CARL" is barely even equal, but at least it's not a Berlin ; ) If black gets the move order wrong, it can get ugly. 

5.0-0 a6 6.Ba4 b5 7.Bb3 d6 8.Ng5 for instance.

Play can also go 5...Nxe4, grabbing both pawns and entering a Riga without a6. For complicated reasons, the the Bg5 line popularized on chesspub didn't seem to work as well, so the pieces vs rook and pawns ending is probably safer.

Another try is 4...Nxe4 where after 0-0 Nd6 we're back in Berlin waters, so I go for 5.d5, muddying the waters a bit. The following is not forced at all, but I did find a fun sacrifice in 5.d5 Ne7 6.Nxe5 c6 7.0-0! leaving the bishop hanging. After 7...cxb5 8.d6 white is a little better to winning better depending on where the knight hops, due to Nxf7 and a nice attack coming. Engines like giving back the piece with 8...Qb6 instead with rough equality, but not so easy to find OTB.