  We discuss and analyze the properties of Casimir forces acting between
nonreciprocal objects in thermal equilibrium. By starting from the
fluctuation-dissipation theorem and splitting the force into those arising from
individual sources, we show that if all temperatures are equal, the resulting
force is reciprocal and is derivable as the gradient of a Casimir (free)
energy. While the expression for the free energy is identical to the one for
reciprocal objects, there are several distinct features: To leading order in
reflections, the free energy can be decomposed as the sum of two terms, the
first corresponding to two reciprocal objects, and the second corresponding to
two anti-reciprocal objects. The first term is negative and typically yields
attraction, while the second can have either sign. For the case of two objects
that are each other's mirror images, the second term is positive and yields
repulsion. The sum of terms can lead to overall repulsive forces, in agreement
with previous observations. Stable configurations, ruled out for reciprocal
cases, appear possible for nonreciprocal objects. We show that for three
objects, a three-body free energy exists, indicating that previously found
persistent heat currents in situations of three objects cannot be used to
produce persistent torques.
