  In 1976, Loupekine introduced (via Isaacs) a very general way of constructing
new snarks from old snarks by cyclically connecting multipoles constructed from
smaller snarks. In this paper, we generalize Loupekine's construction to
produce a variety of snarks which can be drawn with $m$-fold rotational
symmetry for $m\geq 3$ (and often, $m$ odd), constructed as $\mathbb{Z}_{m}$
lifts of \emph{voltage graphs} with certain properties; we call these snarks
\emph{cyclic pseudo-Loupekine snarks}. In particular, we discuss three infinite
families of snarks which can be drawn with $\mathbb{Z}_{m}$ rotational symmetry
whose smallest element is constructed from 3 snarks with 3-fold rotational
symmetry on 28 vertices; one family has the property that the oddness of the
family increases with $m$. We also develop a new infinite family of snarks, of
order $12m$ for each odd $m\geq 3$, which can be drawn with $m$-fold rotational
symmetry and which are constructed beginning with a 3-edge-colorable graph,
instead of a snark.
