  For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$
elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $\sigma$
of its ground set such that every $s-1$ consecutive elements of $\sigma$ are
contained in an $s$-element circuit and every $t-1$ consecutive elements of
$\sigma$ are contained in a $t$-element cocircuit. In the case $s=t$, nearly
$(s, s)$-cyclic matroids have been studied previously. In this paper, we show
that if $M$ is nearly $(s, t)$-cyclic and $n$ is sufficiently large, then these
$s$-element circuits and $t$-element cocircuits are consecutive in $\sigma$ in
a prescribed way, that is, $M$ is "$(s, t)$-cyclic". Furthermore, we show that,
given $s$ and $t$ where $t\ge s$, every $(s, t)$-cyclic matroid on $n > s+t-2$
elements is a weak-map image of the $\left(\frac{t-s}{2}\right)$-th truncation
of a certain $(s, s)$-cyclic matroid. If $s=3$, this certain matroid is the
rank-$\frac{n}{2}$ whirl, and if $s=4$, this certain matroid is the
rank-$\frac{n}{2}$ free swirl.

Combinatorics