  The helicity of a vector field is a measure of the average linking of pairs
of integral curves of the field. Computed by a six-dimensional integral, it is
widely useful in the physics of fluids. For a divergence-free field tangent to
the boundary of a domain in 3-space, helicity is known to be invariant under
volume-preserving diffeomorphisms of the domain that are homotopic to the
identity. We give a new construction of helicity for closed $(k+1)$-forms on a
domain in $(2k+1)$-space that vanish when pulled back to the boundary of the
domain. Our construction expresses helicity in terms of a cohomology class
represented by the form when pulled back to the compactified configuration
space of pairs of points in the domain. We show that our definition is
equivalent to the standard one. We use our construction to give a new formula
for computing helicity by a four-dimensional integral. We provide a Biot-Savart
operator that computes a primitive for such forms; utilizing it, we obtain
another formula for helicity. As a main result, we find a general formula for
how much the value of helicity changes when the form is pushed forward by a
diffeomorphism of the domain; it relies upon understanding the effect of the
diffeomorphism on the homology of the domain and the de Rham cohomology class
represented by the form. Our formula allows us to classify the
helicity-preserving diffeomorphisms on a given domain, finding new
helicity-preserving diffeomorphisms on the two-holed solid torus, and proving
that there are no new helicity-preserving diffeomorphisms on the standard solid
torus. We conclude by defining helicities for forms on submanifolds of
Euclidean space. In addition, we provide a detailed exposition of some standard
`folk' theorems about the cohomology of the boundary of domains in $\R^{2k+1}$.
