  Some new properties of symmetries that disappear as point symmetries after
the first reduction of order of an ODE and reappear after the second are
analyzed from the aspect of three-dimensional subalgebra of symmetries of
differential equations. The form of a hidden symmetry is shown to consist of
two parts, one of which always remains preserved as a point symmetry, and the
second (fundamental) part which behaves as the complete hidden symmetry.
Symmetry that disappears as point symmetry and remains hidden (non-local)
during $n$ reductions of order before reappearing as a point symmetry is also
introduced and termed convertible symmetry of order $n-1.$ We discuss the
necessity for such classification in order to distinguish them from hidden
symmetries of type I and type II, which are defined with respect to reduction
of order by one.
