  It is a commonplace attributed to Kretschmann that any local physical theory
can be represented in arbitrary coordinates using tensor calculus. But the
literature also claims that spinors _as such_ cannot be represented in
coordinates in a curved space-time. These commonplaces are inconsistent, so
what is general covariance for fermions? In fact both commonplaces are wrong.
Ogievetsky and Polubarinov (OP) constructed spinors in coordinates in 1965,
enhancing the unity of physics and helping to spawn nonlinear group
representations. Roughly and locally, OP spinors resemble the orthonormal basis
or tetrad formalism in the symmetric gauge, but they are conceptually
self-sufficient and more economical. The tetrad formalism is thus
de-Ockhamized, with six extra components and six compensating gauge symmetries.
As developed nonperturbatively by Bilyalov, OP spinors admit any coordinates at
a point, but 'time' must be listed first; 'time' is defined by an eigenvalue
problem involving the metric and diag(-1,1,1,1), the product of which must have
no negative eigenvalues. Thus even formal general covariance requires
reconsideration; admissible coordinates should be sensitive to the types and
_values_ of the fields. Apart from coordinate order and the usual spinorial
two-valuedness, (densitized) Ogievetsky-Polubarinov spinors form, with the
(conformal part of the) metric, a nonlinear geometric object, with Lie and
covariant derivatives. The mild consequences of coordinate order are explored
for the conventionality of simultaneity in Special Relativity and the
Schwarzschild solution in General Relativity.
