Honda hybrid always has the IC engine running while the vehicle is in motion.
With one exception (two if you count the Fit Hybrid in Japan) but yes. The 2nd gen Accord Hybrid has a 1-speed clutched gearbox from engine to wheels, and it's disconnected a majority of the time, acting only as a generator. The electric motor is responsible for all acceleration. I believe this is the only Honda engine in a production car that uses an Atkinson cycle.
All other Honda hybrids use an electric motor permanently attached to the output shaft which cannot be disengaged from the engine, and the engines use OTTO cycle. The purpose of the electric motor is to allow the downsized engine to feel/behave like a larger engine when accelerating, and consume the fuel and have the losses/drag/waste of a smaller engine when cruising, and it works. Honda's hybrids typically shine on the highway, whereas Toyotas do much better in the city. The first-gen Civic Hybrid was easily capable of 50-60mpg at highway speeds and the G1 Insight could approach 100mpg with the A/C off, but both suffered in stop and go traffic due to the engine always needing to run.
It's worth mentioning that most newer Honda hybrids DO have EV modes, but they still turn the crankshaft even when the engine isn't firing, which IMO is a poor design. The VTEC mechanism closes all the valves so it isn't pumping any air, but there are still frictional losses. Toyota's transmission is a far more elegant solution, but for whatever reason their lineup isn't capable of delivering the same highway economy.