  We revisit the particle-hole symmetry of the one-dimensional ($D=1$)
fermionic spinless Hubbard model, associating that symmetry to the invariance
of the Helmholtz free energy of the one-dimensional spin-1/2 $XXZ$ Heisenberg
model, under reversal of the longitudinal magnetic field and at any finite
temperature. Upon comparing two regimes of that chain model so that the number
of particles in one regime equals the number of holes in the other, one finds
that, in general, their thermodynamics is similar, but not identical: both
models share the specific heat and entropy functions, but not the internal
energy per site, the first-neighbor correlation functions, and the number of
particles per site. Due to that symmetry, the difference between the
first-neighbor correlation functions is proportional to the $z$-component of
magnetization of the $XXZ$ Heisenberg model. The results presented in this
paper are valid for any value of the interaction strength parameter $V$, which
describes the attractive/null/repulsive interaction of neighboring fermions.
