  Typical heat engines exhibit a kind of homotypy: The heat exchanges between a
cyclic heat engine and its two heat reservoirs abide by the same function type;
the forward and backward flows of an autonomous heat engine also conform to the
same function type. This homotypy mathematically reflects in the existence of
hidden symmetries for heat engines. The heat exchanges between the cyclic heat
engine and its two reservoirs are dual under the joint transformation of parity
inversion and time-reversal operation. Similarly, the forward and backward
flows in the autonomous heat engine are also dual under the parity inversion.
With the consideration of these hidden symmetries, we derive a generic
nonlinear constitutive relation up to the quadratic order for tight-coupling
cyclic heat engines and that for tight-coupling autonomous heat engines,
respectively.
