  We show that in a spacetime geometry with general connection, there exist
field redefinitions that replace the mixed-symmetry nonmetricity of the
asymmetric connection with the sum of torsion and the field strength of special
conformal transformations of a Lorentzian connection. In contrast to general
relativity, generic action functionals give different results for metric versus
Palatini-style variations. We consider some of the resulting issues in
Poincar\`e gauge theory, where it is natural to vary both the solder form and
the spin connection, and where the action may differ from the Einstein-Hilbert
case. Independent variation may introduce nonmetricity as well as torsion.
Appending the defining equation of nonmetricity to the Maurer-Cartan equations
of the Poincar\`e Lie algebra, we show that the mixed symmetry components of
nonmetricity can be absorbed into an altered torsion tensor, restoring local
Lorentz symmetry and metric compatibility while including quadratic terms in
the redefined curvature. In order to recover the original torsion and nonmetric
fields, we replace the definition of nonmetricity by an additional structure
equation. We show that the maximal Lie algebra compatible with the enlarged set
is isomorphic to the conformal Lie algebra. Therefore, in a Lorentzian
conformal geometry the sum of torsion and the field strength of special
conformal transformations is given by the mixed symmetry nonmetricity of an
equivalent asymmetric system.
