Herein we consider material tensors that pertain to thin sheets or thin films, which we model as two-dimensional objects. We assume that the thin sheet in question carries a crystallographic texture characterized by an orientation distribution function defined on the rotation group SO(3), which is almost transversely-isotropic about the sheet normal so that mechanical and physical properties of the thin sheet have weak planar-anisotropy. We present a procedure by which a special orthonormal basis can be determined in each tensor subspace invariant under the action of the orthogonal group O(2). We call members of such special bases irreducible basis tensors under O(2). For the class of thin sheets in question, we derive a representation formula in which each tensor in any given tensor subspace Z is written as the sum of a transversely-isotropic term and a linear combination of orthonormal irreducible basis tensors in Z, where the coefficients are given explicitly in terms of texture coefficients and undetermined material parameters. In addition to the general representation formula, we present also the specialized form for subspaces of tensor products of second-order symmetric tensors, a type commonly found in mechanics of materials.