We present a description of isochronous centres of planar vector
fields $X$ by means of their groups of symmetries. More precisely,
given a normalizer $U$ of $X$ (i.e., $[X,U]=\mu X$, where $\mu$
is a scalar function), we provide a necessary and sufficient
isochronicity condition based on $\mu$. This criterion extends the
result of Sabatini and Villarini that establishes the equivalence
between isochronicity and the existence of commutators ($[X,U]=
0$). We put also special emphasis on the mechanical aspects of
isochronicity; this point of view forces a deeper insight into the
potential and quadratic-like Hamiltonian systems. For these
families we provide new ways to find isochronous centres,
alternative to those already known from the literature.