I've been pondering this myself. In aircraft structural design and repair, carbon fiber is used as a replacement for aluminum. To attain stiffness, fibers must be oriented 90 degrees to the bending force. So on a spar, the fibers run the length of the spar. To carry twisting loads, as seen on surfaces like wings and flaps, bidirectional cloth is used. These cloths are available from the same distributors but it's probable that sufficient layers to obtain the required stiffness would negate any weight savings.
It's possible to test the difference in stiffness. Build 2 faux flaps, maybe 24" X 2" (over-large to make measurement easier), one covered in carbon fiber and the other in a traditional material. Mount them on a piece of wood the same as they would on an aircraft with hinges. Lock one end in place where the control horn would be and attach a spring scale to the opposite trailing edge. Measure the force required to twist the flap perhaps 1/4". You should be able to see some difference if there is a difference in stiffness.