  In this paper, we revisit the claim that many partition functions are
invariant under reflecting temperatures to negative values (T-reflection). The
goal of this paper is to demarcate which partition functions should be
invariant under T-reflection, and why. Our main claim is that
finite-temperature path integrals for quantum field theories (QFTs) should be
T-reflection invariant. Because multi-particle partition functions are equal to
Euclidean path integrals for QFTs, we expect them to be T-reflection invariant.
Single-particle partition functions though are often not invariant under
T-reflection. Several exactly solvable systems are non-invariant under naive
T-reflection, but are likely invariant under an extended T-reflection. We give
example systems that are T-reflection invariant but are (1) non-unitary, (2)
chiral, (3) interacting, (4) non-supersymmetric, or (5) non-conformal, and (6)
argue that T-reflection is unrelated to time-reversal. Finally, we study the
interplay between T-reflection and perturbation theory in the anharmonic
harmonic oscillator in quantum mechanics and in Yang-Mills in four-dimensions.
This is the first in a series of papers on temperature-reflections.
