What you describe is a half-bridge. Two half-bridges is an
H-bridge, which you would want if you need to have a
continuous, proportional reverse (otherwise, half bridge and
an F/R DPDT switch might do). If you truly need a full (H)
bridge then you really want to look for a full bridge driver
as there may be better control of cross-leg dead times and
such, than you can get with separate drivers.
Also many of the high voltage drivers using NMOS high side
switches, have maximum high side "on" times because they
are bootstrapped. If you are working at lower voltages and
need to be able to run all the way to 100% duty cycle,
you may prefer a bridge driver that uses NMOSlow side and
PMOS high side, despite the slightly higher cost of equal-
current/voltage PMOSFETs.