  We study transformations of the dynamical fields - a metric, a flat affine
connection and a scalar field - in scalar-teleparallel gravity theories. The
theories we study belong either to the general teleparallel setting, where no
further condition besides vanishing curvature is imposed on the affine
connection, or the symmetric or metric teleparallel gravity, where one also
imposes vanishing torsion or nonmetricity, respectively. For each of these
three settings, we find a general class of scalar-teleparallel action
functionals which retain their form under the aforementioned field
transformations. This is achieved by generalizing the constraint of vanishing
torsion or nonmetricity to non-vanishing, but algebraically constrained torsion
or nonmetricity. We find a number of invariant quantities which characterize
these theories independently of the choice of field variables, and relate these
invariants to analogues of the conformal frames known from scalar-curvature
gravity. Using these invariants, we are able to identify a number of physically
relevant subclasses of scalar-teleparallel theories. We also generalize our
results to multiple scalar fields, and speculate on further extended theories
with non-vanishing, but algebraically constrained curvature.
