It seems to me that the obligation to housework that you and many other women feel is tied to the linguistic structure is being repeated in the way we talk about it. It is interesting how that obligation has become culturally associated with our patterns of talk.

I agree that it does not matter grammatically to entitle the picture as "Greg and Me" or "Greg and I." To take this conversation to a different level, I think I'd make the decision based on how I would want to be looked at, as either a subject or an object.