Why does the DHC-6 pitch down when the flaps are retracted, rather than up?
The Twin Otter isn't unique.  Flaps have varying effects on different airplanes.  Some pitch up with flaps extended, some pitch down.   The pitching moment from flaps isn't just from the nose down moment from the increase in airfoil camber; it includes effects of increased downwash on the tail, which creates a pitch up moment, and to some degree, changes in the vertical aerodynamic center resulting from adding drag up high or down low (like the mild pitch down moment caused by gear extension).  

In the Twin Otter the double-slotted flaps are extremely powerful and create a very strong downwash increase at high angles as opposed to just mostly turbulence you get with simpler flap setups. The result is the net effect of flap extension, with the high wing and the location of the horizontal stabilizer, is pitch up because those forces are stronger than the nose down moment created at the wing itself by the camber change.