Many planar textiles exhibit fabric shear, the rotation of threads relative to each other, which imbues textiles with additional compliance in the direction of their so-called true bias. Since antiquity, designers have utilized this deformation mode to produce fabrics that stretch, drape, and conform to curved surfaces. This work introduces Pantographic Interwoven Composable Lattices (“PICLs”), a novel family of three-dimensional interwoven metamaterials. PICLs exhibit a single s zero-mode analogous to fabric shear, mediated by the reconfiguration of interlocking discrete planar elements. First a procedural design method is introduced which generalizes a graph embedding approach to three-dimensional complexes [1]. This method enables a PICL structure’s bias direction to be varied programmatically, enabling architectures with locally varied anisotropic stiffness. The kinematics of the PICL’s internal reconfiguration are shown to be well-represented by a parameterized family of zero-energy deformation gradients, F(θ), derived from the local constraints of the interlocking elements. Further, the onset of self-locking, caused by self-contact, is shown to be directly influenced by the volumetric density, analogous to the jamming transition observed by Zhou et al for polycatenated structures [2]. To understand the mechanical properties of PICLs, a homogenized continuum model based on second-gradient energies is proposed, derived from [3], which aims to capture the influence of both planar stresses and out-of-plane bending on elastic response. PICLs are additionally shown to be spatially composable: separate PICL regions with differing bias directions may be interwoven at boundaries, such that their individual zero-modes are coupled into a single structural degree of freedom. This composability enables PICLs to serve as building blocks for structurally-embodied kinematic chains capable of executing high-amplitude, nonlinear transformations without the need for discrete joints. PICL models are fabricated via polymer-based additive manufacturing approaches, enabling validation of the mechanical model and demonstrating potential uses in programmable metamaterials and damage-resilient soft robotics.