  A linear section of a double vector bundle is a parallel pair of sections
which form a vector bundle morphism; examples include the complete lifts of
vector fields to tangent bundles and the horizontal lifts arising from a
connection in a vector bundle. A grid in a double vector bundle consists of two
linear sections, one in each structure, and thus provides two paths from the
base manifold to the top space; the warp of the grid measures the lack of
commutativity of the two paths.
  For a double vector bundle $D$, a linear section induces a linear map on the
relevant dual of $D$ and thence a linear section of the iterated dual; we call
this the corresponding squarecap section. This notion makes precise the
relationships between various standard $1$-forms and vector fields that are not
dual in the usual sense. A grid in $D$ induces squarecap sections in the two
iterated duals. These iterated duals are themselves in duality and we obtain an
expression for the warp of a grid in terms of a pairing. This result extends
the relationships between various standard formulas, and clarifies the iterated
dualization involved in the concept of double Lie algebroid.
  In the first part of the paper we review the nonstandard pairing of the two
duals of a double vector bundle, showing in particular that it arises as a
direct generalization of the Legendre-type antisymplectomorphism $T^*(A^*)\to
T^*(A)$ for any vector bundle $A$ of Mackenzie and Xu.
