  We elucidate the vector space (twisted relative cohomology) that is
Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology)
in general spacetime dimension. The pairing between these spaces - an algebraic
invariant called the intersection number - extracts integral coefficients for a
minimal basis, bypassing the generation of integration-by-parts identities.
Dual forms turn out to be much simpler than their Feynman counterparts: they
are supported on maximal cuts of various sub-topologies (boundaries). Thus,
they provide a systematic approach to generalized unitarity, the reconstruction
of amplitudes from on-shell data. In this paper, we introduce the idea of dual
forms and study their mathematical structures. As an application, we derive
compact differential equations satisfied by arbitrary one-loop integrals in
non-integer spacetime dimension. A second paper of this series will detail
intersection pairings and their use to extract integral coefficients.
