Only one person in a household can file as head of household. One of the requirements for head of household filing status is that you paid more than half of the cost of keeping up a home for the year. Only one of you can pay more than half of the cost of keeping up the home. If one paid more than half, the other must have paid less than half.
If you really split all the household expenses exactly equally, then neither of you can file as head of household, since neither of you paid more than half of the cost of keeping up the home.
