In the context of orchestral trumpet playing, "rotary" refers to a specific style and design of trumpet. Rotary trumpets (named for a particular type of valve construction) are typically used in German and Austrian orchestras rather than the piston trumpets we use in the U.S. They look and are played a little differently than "normal" piston trumpets. In fact, audience members often ask us why we're holding our instruments sideways.
Chris Still, of the LA Phil trumpet section, shows off a rotary trumpet, which is normally used in German and Austrian orchestras and thus was the perfect instrument to bring along on this tour.