  We investigate odd-in-time - or odd-frequency - pairing of fermions in
equilibrium systems within the particle-number-conserving framework of Penrose,
Onsager and Yang, where superfluid order is defined by macroscopic eigenvalues
of reduced density matrices. We show that odd-frequency pair correlations are
synonymous with even fermion-exchange symmetry in a time-dependent correlation
function that generalises the two-body reduced density matrix. Macroscopic
even-under-fermion-exchange pairing is found to emerge from conventional
Penrose-Onsager-Yang condensation in two-body or higher-order reduced density
matrices through the symmetry-mixing properties of the Hamiltonian. We identify
and characterise a transformer matrix responsible for producing macroscopic
even fermion-exchange correlations that coexist with a conventional Cooper-pair
condensate, while a generator matrix is shown to be responsible for creating
macroscopic even fermion-exchange correlations from hidden orders such as a
multi-particle condensate. The transformer scenario is illustrated using the
spin-balanced s-wave superfluid with Zeeman splitting as an example. The
generator scenario is demonstrated by the composite-boson condensate arising
for itinerant electrons coupled to magnetic excitations. Structural analysis of
the transformer and generator matrices is shown to provide general conditions
for odd-frequency pairing order to arise in a given system. Our formalism
facilitates a fully general derivation of the Meissner effect for odd-frequency
superconductors that holds also beyond the regime of validity for mean-field
theory.
