Matroids with a cyclic arrangement of circuits and cocircuits

For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $sigma$ such that every set of $t-1$ consecutive elements in $sigma$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $sigma$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element circuits and cocircuits appear in swirls. Furthermore, we show that any appropriate concatenation $Phi$ of $sigma$ is a flower. If $t$ is odd, then $Phi$ is a daisy, but if $t$ is even, then, depending on $M$, it is possible for $Phi$ to be either an anemone or a daisy.