Does a quadcopter have flight modes beyond traditional aircraft?
As DeltaLima noted, pretty much all flight modes of a quadcopter are "non-traditional" -- they're a relatively new aircraft design.

Most quadcopter designs lack anything resembling "wings", so they can't "fly" in an "X-Wing" orientation (with the rotors producing entirely lateral thrust) -- at least not as we traditionally define flight.

They can, as you surmised, do a climb-and-run maneuver in that orientation: Gain a lot of altitude by using vertical thrust, tilt the platform near-90-degrees in the direction you want to travel (so your thrust is pretty much lateral), and accelerate in that direction while rapidly losing altitude.  

I've no idea if this is more or less efficient than the usual slight tilt they fly with, but it's certainly less stable since you're in vertical free-fall and would need to redirect thrust to be vertical again with enough time/altitude to arrest the descent before an ungraceful impact with the ground.  


  X-Wings work so well because the microgravity environment of deep space isn't trying to drag you back to an impact with the ground - it wouldn't work well deep in a planet's gravity well, which is where most of our aircraft operate.
  Of course if you had access to a microgravity environment with an atmosphere you could configure a quadcopter to fly like an X-Wing fighter…




The quadcopter's thrust configuration does open up the option for some interesting maneuvers though.

It's theoretically possible to "flip" a quadcopter (around at least 8 axes) if you have sufficient altitude (especially if the rotors can be reversed in flight to assist in the maneuvers).  

It's also possible to "rock the wings" (really rocking the whole platform) in many directions by alternating thrust on pairs of rotors.