then I will take your hands expecting to be the lady while if you instead put them as:

then I will expect to be the gent. On the other hand, the cross hand grip is symmetrical:

Two people with their hands in a symmetrical grip might or might not agree on which roles they are. So for a symmetrical grip, the convention is that the person who offers the grip is leading. Someone offering their hand in this way, then, is indicating that they would like to be the gent. This is a little different than the two hand grips; with those it doesn't matter who sticks their hands out first.

(I find being really consistent about hand positions—up for gent, down for lady—is very helpful in keeping track of which role I'm dancing.)