The Hamilton-Waterloo Problem HWP$(v;m,n;α,β)$ 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 $α$ $C_m$-factors and $β$ $C_n$-factors, where $3 \leq m < n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1 \in \{α,β\}$ or when $α$ and $β$ 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$(v;2m,2n;α,β)$ for odd $α$ and $β$ whenever the obvious necessary conditions hold, except possibly if $β=1$; $β=3$ and $\gcd(m,n)=1$; $α=1$; or $v=2mn/\gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above.