The Hex Inverter is only needed if you want to simplify a H-bridge using all n-chan MOSFETS.
Ok, if you use P-chan --- N-chan complementary pairs, you connect the gates together and you turn one (pair) side on, after the other comp. pair has been turned off with a simple inverter(npn transistor) to provide the communtating.
With all N-chans, you need to have separate drive signals for each gate. The top N-chan must be off when it's bottom brother is on, on one side of the bridge, and we have the opposite control signals on the other side of the bridge....can't you figure out how to do this? It means that that only one upper MOSFET gate of each side of the bridge is turned on, and the lower MOSFET is turned-on on the other side of the bridge. And so it alternates back and forth.