Although solid helium-4 ( 4He) may be a supersolid, it also exhibits many phenomena unexpected in that context. We studied relaxation dynamics in the resonance frequency f(T) and dissipation D(T) of a torsional oscillator containing solid 4He. With the appearance of the "supersolid" state, the relaxation times within f(T) and D(T) began to increase rapidly together. More importantly, the relaxation processes in both D(T) and a component of f(T) exhibited a complex synchronized ultraslow evolution toward equilibrium. Analysis using a generalized rotational susceptibility revealed that, while exhibiting these apparently glassy dynamics, the phenomena were quantitatively inconsistent with a simple excitation freeze-out transition because the variation in f was far too large. One possibility is that amorphous solid 4He represents a new form of supersolid in which dynamical excitations within the solid control the superfluid phase stiffness. © 2009 by the American Association for the Advancement of Science.