[math-fun] church of the sub-genus?

This is doubtless crankery, but... In computing the Euler characteristic I notice that handles get twice the weight of cross-caps. Plugging p=1/2 in for the "genus" in the chromatic formula floor((7+sqrt(1+48*p))/2) gives 6, the number for the Mobius strip. Does that figure, or is it just an accident? Is there some general chromatic expression that covers non-orientable surfaces? If we can have "fractional dimension", can we somehow have "fractional genii"? Happy Holey-days!