  The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a
2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with
the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$
$C_n$-factors, where $3 \leq m < n$. In the case that $m$ and $n$ are both
even, the problem has been solved except possibly when $1 \in \{\alpha,\beta\}$
or when $\alpha$ and $\beta$ are both odd, in which case necessarily $v \equiv
2 \pmod{4}$. In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP$(v;2m,2n;\alpha,\beta)$ for odd $\alpha$ and $\beta$ whenever the obvious
necessary conditions hold, except possibly if $\beta=1$; $\beta=3$ and
$\gcd(m,n)=1$; $\alpha=1$; or $v=2mn/\gcd(m,n)$. This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above.
