  This paper studies relationships between the order reductions of ordinary
differential equations derived by the existence of $\lambda$-symmetries,
telescopic vector fields and some nonlocal symmetries obtained by embedding the
equation in an auxiliary system. The results let us connect such nonlocal
symmetries with approaches that had been previously introduced: the exponential
vector fields and the $\lambda$-coverings method. The $\lambda$-symmetry
approach let us characterize the nonlocal symmetries that are useful to reduce
the order and provides an alternative method of computation that involves less
unknowns. The notion of equivalent $\lambda$-symmetries is used to decide
whether or not reductions associated to two nonlocal symmetries are strictly
different.
