I have long believed that a happy partnership means that both partners have to contribute fairly equally to the household and that no one should be left with the worst job of all: monitoring the other's contribution.
To make this possible I think a fairly clear division of labour can help to reduce the amount of boring negotiations over household chores.
This is roughly how we have come to divide ours (for now, obviously things will change as kids grow etc):
Cooking & food prep
Gift planning & buying
Daytime clean up (including dishes - although he often does breakfast ones before work)
Schedules (ours & kids)
Charlie nighttime parenting
Weekday at home parenting work
Evening clean up (including dishes)
Mail sorting, dealing with contents
Lily nighttime parenting, including story & bedtime
Driving (when we're together)
Holiday planning & booking
Laundry (but he does far more than me)
Farmers market shopping
Of course sometimes he cooks or shops for food, and sometimes I clean up in the evening or do the banking, but this is the general pattern.
Do you have a pattern at your place or do you tend to alternate more?