I'm talking about movement left and right without changing attitude. This is a motion that can be done with a quadcopter and to a limited extent a helicopter (I presume).
On a helicopter a left/right movement of the cyclic will command a roll motion from the rotor, but in a hover the aircraft would only roll slightly then begin to move sideways. Does the pilot just call this a roll or is there another, more appropriate word?
I thought of slip or crab, but those describe specific attitudes associated with forward flight. They don't seem to fit the bill here.
In our NATOPS manuals, and in our trained procedures in both Search and Rescue ops, and cargo ops, in the Navy the crewman would call for us to "slide left" or "slide right" as he positioned us over the pick up point. (Easy Left/Easy Right as we got closer to the spot).
The FAA term "sideward hovering" suffices, but it is two words. If you want one word, slide is about right.
The answer is in the question: lateral, fwd/aft is longitudinal. The cyclic stick directions are cyclic longitudinal and cyclic lateral: move the stick and the rotor tilts, producing direct translational movement. The helicopter fuselage follows the rotor angle and therefore pitches/rolls as well, but more as a (desired) side effect. Desired because they enable instant judgement on the amount of stick applied. In a fixed wing, stick fwd/sideways is pitch stick and roll stick, because stick inputs primarily affect the aircraft attitudes. Longitudinal and lateral are helicopter terms because stick inputs (in a hover) mainly affect translations. And yes on a 6-DoF simulator motion system they are called Surge and Sway.
Sideward hovering flight may be necessary to move the helicopter to a specific area when conditions make it impossible to use forward flight. During the maneuver, a constant groundspeed, altitude, and heading should be maintained.
There are also the forward and rearward hovers, and the hovering turn. These hovers are probably all specific to helicopters, where straight and level flight is designed to be performed in the forward direction. Aircraft like quadcopters are less directional and can generally fly "straight and level" in any direction relative to the quadcopter.