Hi. I live on the UES and have seen what you're talking about... at least, I think I have. I am not familiar with departures flying southwest over Manhattan, so I went to airnav and looked at KLGA's departure procedures. I have attached them below. It seems the RWY 22 departure climbs to the DME, then turns left to 070. So, unless another procedure is in place for 22, I don't believe this is the RWY being used. However, it seems that the Maspeth climb from RWY 13 may be what you are seeing. However, this ends up flying over Manhattan at a heading of 340. I'm a little unsure if this is correct, but for departures this seems to be the best bet.
As far as approaches go, I have definitely seen many planes fly south (arriving from the North/East?) over Manhattan and then turn left, head up the east river, and turn right for the Expressway Visual approach into 31. Maybe this is what you're seeing?
Any other opinions from people who actually know this for sure??