I would say we are equals, but with different rolls when it comes to household responsibilities. My DH is retired, so has all day at home. He does most of the housekeeping and laundry, and a good bit of cooking, plus all the outside chores.
That makes me sound like I don't do anything, but I work full time still, and do all the 'paperwork' and anything that involves a phone call. He admittedly hates talking to people on the phone, other than to chat with friends or family. So today my list is to call to check up on a new door ordered months ago, stop to pick up some cash at the bank, stop to pick up new contact lenses, and organize the bills and mail after having been away for a few weeks.
He does his list his way, and if I am fussy, then I can do it. He puts everything in the washer and dryer, so if I want a few items on delicate or hung up to dry, then I save them for a special load and do it myself. He washes floors, but always forgets corners or baseboards, so I will do that, plus dusting of light fixtures, etc.; more detailed cleaning.
We don't really have any conflicts, not major ones anyway. He doesn't like how I organize the fridge, but I hate how he leaves water in the sink if he is rinsing some dishes. Minor issues.
I also have control of the shopping buggy!