No, then engines do not change their direction of rotation at all, at any time.
In a turboprop such as the one I fly, going to reverse means reducing the pitch of the propeller blades through flat pitch so that they generate thrust opposite the direction of travel of the airplane. It's a controllable pitch prop, similar to what you would see on a 172RG or an Arrow, we just have a much wider pitch envelope to work with.
In jets, forward engine thrust is being deflected. Jet thrust reversers (TRs) can be divided in to two categories ... pre-exit and post-exit. Post-exit reversers, such as those found on airplanes like the MD-80, DC-9, Beechjet 400A, etc., are basically "clamshell doors" which open behind the engine and deflect the exhaust forward to help slow the airplane. Pre-exit reversers change the direction of the air before it leaves the exhaust. In most high-bypass-turbofan engines, only the air in the fan section is reversed. This is what occurs on CFM-powered 737s (the -300 and later series). A "translating sleeve", as it is called, slides backwards on the engine cowl. This causes a panel to deflect the front-to-back air flow through the fan section forward, where it is exhausted through that grid you see from your seat when the sleeve opens. On the Airbus it is similar (many Airbus 320 family aircraft are powered by the same CFM56 engines as on the 737), except that rather than a translating sleeve, there are those series of doors that open to direct engine thrust opposite the direction of travel.
I hope this makes more sense now.
FL270