  By breaking the spin symmetry in the relativistic domain, a powerful tool in
physical sciences was lost. In this work, we examine an alternative of spin
symmetry for systems described by the many-electron Dirac-Coulomb Hamiltonian.
We show that the square of many-electron operator $\mathcal{K}_+$, defined as a
sum of individual single-electron time-reversal (TR) operators, is a linear
Hermitian operator which commutes with the Dirac-Coulomb Hamiltonian in a
finite Fock subspace. In contrast to the square of a standard unitary
many-electron TR operator $\mathcal{K}$, the $\mathcal{K}^2_+$ has a rich
eigenspectrum having potential to substitute spin symmetry in the relativistic
domain. We demonstrate that $\mathcal{K}_+$ is connected to $\mathcal{K}$
through an exponential mapping, in the same way as spin operators are mapped to
the spin rotational group. Consequently, we call $\mathcal{K}_+$ the generator
of the many-electron TR symmetry. By diagonalizing the operator
$\mathcal{K}^2_+$ in the basis of Kramers-restricted Slater determinants, we
introduce the relativistic variant of configuration state functions (CSF),
denoted as Kramers CSF. A new quantum number associated with $\mathcal{K}^2_+$
has potential to be used in many areas, for instance, (a) to design effective
spin Hamiltonians for electron spin resonance spectroscopy of heavy-element
containing systems; (b) to increase efficiency of methods for the solution of
many-electron problems in relativistic computational chemistry and physics; (c)
to define Kramers contamination in unrestricted density functional and
Hartree--Fock theory as a relativistic analog of the spin contamination in the
nonrelativistic domain.
