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):
Me:
Meal planning
Cooking & food prep
Food shopping
Gift planning & buying
Daytime clean up (including dishes - although he often does breakfast ones before work)
Schedules (ours & kids)
Breastfeeding
Charlie nighttime parenting
Weekday at home parenting work
Him:
Evening clean up (including dishes)
Paid work
Bill paying
Banking
Mail sorting, dealing with contents
Lily nighttime parenting, including story & bedtime
Driving (when we're together)
IT troubleshooting
Both:
Gardening
Constant tidying
Holiday planning & booking
Parenting
Bathroom cleaning
Laundry (but he does far more than me)
Household shopping
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?