  We investigate the behavior under Lorentz tranformations of perturbative
coefficient functions in a collinear twist-3 formalism relevant for high-energy
observables including transverse polarization of hadrons. We argue that those
perturbative coefficient functions can, {\it a priori}, acquire quite different
yet Lorentz-invariant forms in various frames. This somewhat surprising
difference can be traced back to a general dependence of the perturbative
coefficient functions on lightcone vectors which are introduced by the twist-3
factorization formulae and which are frame-dependent. One can remove this
spurious frame dependence by invoking so-called Lorentz invariance relations
(LIRs) between twist-3 parton correlation functions. Some of those relations
for twist-3 distribution functions were discussed in the literature before. In
this paper we derive the corresponding LIRs for twist-3 fragmentation
functions. We explicitly demonstrate that these LIRs remove the lightcone
vector dependence by considering transverse spin observables in the
single-inclusive production of hadrons in lepton-nucleon collisions, $\ell N\to
hX$. With the LIRs in hand, we also show that twist-3 observables in general
can be written solely in terms of three-parton correlation functions.
