  We study a single-valued integration pairing between differential forms and
dual differential forms which subsumes some classical constructions in
mathematics and physics. It can be interpreted as a $p$-adic period pairing at
the infinite prime.
  The single-valued integration pairing is defined by transporting the action
of complex conjugation from singular to de Rham cohomology via the comparison
isomorphism. We show how quite general families of period integrals admit
canonical single-valued versions and prove some general formulas for them. This
implies an elementary 'double copy' formula expressing certain singular volume
integrals over the complex points of a smooth projective variety as a quadratic
expression in ordinary period integrals of half the dimension. We provide
several examples, including non-holomorphic modular forms, archimedean
N\'{e}ron-Tate heights on curves, single-valued multiple zeta values and
polylogarithms.
  In a sequel to this paper we apply this formalism to the moduli space of
curves of genus zero with marked points, to deduce a recent conjecture due to
Stieberger in string perturbation theory, which states that closed string
amplitudes are the single-valued projections of open string amplitudes.
