  Microscopic symmetries impose strong constraints on the elasticity of a
crystalline solid. In addition to the usual spatial symmetries captured by the
tensorial character of the elastic tensor, hidden non-spatial symmetries can
occur microscopically in special classes of mechanical structures. Examples of
such non-spatial symmetries occur in families of mechanical metamaterials where
a duality transformation relates pairs of different configurations. We show on
general grounds how the existence of non-spatial symmetries further constrains
the elastic tensor, reducing the number of independent moduli. In systems
exhibiting a duality transformation, the resulting constraints on the number of
moduli are particularly stringent at the self-dual point but persist even away
from it, in a way reminiscent of critical phenomena.
