  We study $SU(2)$ calorons, also known as periodic instantons, and consider
invariance under isometries of $S^1\times\mathbb{R}^3$ coupled with a
non-spatial isometry called the rotation map. In particular, we investigate the
fixed points under various cyclic symmetry groups. Our approach utilises a
construction akin to the ADHM construction of instantons -- what we call the
monad matrix data for calorons -- derived from the work of Charbonneau and
Hurtubise. To conclude, we present an example of how investigating these
symmetry groups can help to construct new calorons by deriving Nahm data in the
case of charge $2$.
