  In this Letter we consider a general quadratic parity-preserving theory for a
general flat connection. Imposing a local symmetry under the general linear
group singles out the general teleparallel equivalent of General Relativity
carrying both torsion and non-metricity. We provide a detailed discussion on
the teleparallel equivalents of General Relativity and how the two known
equivalents, formulated on Weitzenb\"ock and symmetric teleparallel geometries
respectively, can be interpreted as two gauge-fixed versions of the general
teleparallel equivalent. We then explore the viability of the general quadratic
theory by studying the spectrum around Minkowski. The linear theory generally
contains two symmetric rank-2 fields plus a 2-form and, consequently, extra
gauge symmetries are required to obtain potentially viable theories.
