  Compactified Yang-Mills theories with one universal extra dimension were
found [arXiv:1008.4638] to exhibit two types of gauge invariances: the standard
gauge transformations (SGTs) and the nonstandard gauge transformations (NSGTs).
In the present work we show that these transformations are not exclusive to
compactified scenarios. Introducing a notion of hidden symmetry, based on the
fundamental concept of canonical transformation, we analyse three different
gauge systems, each of which is mapped to a certain effective theory that is
invariant under the so-called SGTs and NSGTs. The systems under discussion are:
(i) four dimensional pure $SU(3)$ Yang-Mills theory, (ii) four dimensional
$SU(3)$ Yang-Mills with spontaneous symmetry breaking, and (iii) pure
Yang-Mills theory with one universal compact extra dimension. The canonical
transformation, that induces the notion of hidden symmetry, maps objects with
well defined transformation laws under a gauge group $G$ to well defined
objects under a non-trivial subgroup $H \subset G$. In the case where
spontaneous symmetry breaking is present, the set of SGTs corresponds to the
group into which the original gauge group is broken into, whereas the NSGTs are
associated to the broken generators and can be used to define the unitary
gauge. For the system (iii), the SGTs coincide with the gauge group $
SU(N,\cal{M}^{4}) $, whereas the NSGTs do not form a group; in this system the
'fundamental' theory and the effective one are shown to be classically
equivalent.
