  Some key features of the symmetries of the Schr\"odinger equation that are
common to a much broader class of dynamical systems (some under construction)
are illustrated. I discuss the algebra/superalgebra duality involving first and
second-order differential operators. It provides different viewpoints for the
spectrum-generating subalgebras. The representation-dependent notion of
on-shell symmetry is introduced. The difference in associating the
time-derivative symmetry operator with either a root or a Cartan generator of
the $sl(2)$ subalgebra is discussed. In application to one-dimensional
Lagrangian superconformal sigma-models it implies superconformal actions which
are either supersymmetric or non-supersymmetric.
