  Lorentz violation is a candidate quantum-gravity signal, and the
Standard-Model Extension (SME) is a widely used parametrization of such a
violation. In the gravitational SME sector, there is an elusive coefficient for
which no effects have been found. This is known as the $t$ puzzle and, to date,
it has no compelling explanation. This paper analyzes whether there is a
fundamental explanation for the $t$ puzzle. To tackle this question, several
approaches are followed. Mainly, redefinitions of the dynamical fields are
studied, showing that other SME coefficients can be moved to nongravitational
sectors. It is also found that the gravity SME sector can be consistently
treated \textit{\`a la} Palatini, and that, in the presence of spacetime
boundaries, it is possible to correct its action to get the desired equations
of motion. Moreover, through a reformulation as a Lanczos-type tensor, some
problematic features of the $t$ term, which should arise at the
phenomenological level, are revealed. The most important conclusion of the
paper is that there is no evidence of a fundamental explanation for the $t$
puzzle, suggesting that it may be linked to the approximations taken at the
phenomenological level.
