Symmetry is usually viewed as a constraint on physical responses, but it can also be used to design the fields that control them. Here, we adopt the representation-matching principle used to identify conjugate fields of order parameters and apply it to response tensors. A linear coupling, the key ingredient for sign-switching control, is symmetry-allowed when a component (or a combination of components) of the control field and the selected response component share the same irreducible representation of the magnetic point group (MPG) of the system. Applied to the magneto-optical Kerr effect, this principle yields a piezo-Kerr effect, in which specific strain combinations activate a Kerr response forbidden in the unstrained state. Enumerating all 122 MPGs, we identify 16 groups in which the unstrained Kerr response is forbidden while a linear strain-induced contribution is allowed. These groups are realized by altermagnets and noncollinear antiferromagnets with symmetry-enforced vanishing net magnetization. The active strain channels and the associated first-order sign reversals follow directly from symmetry. First-principles calculations for bulk MnTe₂ and CoF₂, and Cr₂Se₂O and Cr₂Te₂O monolayers confirm these predictions. In these materials, the Kerr rotation vanishes without strain, whereas 0.5-3% strain produces rotations ranging from tens to hundreds of millidegrees, and reversing the matched strain channel reverses the Kerr sign. These results establish a general route to large and switchable Kerr responses emerging from symmetry-forbidden states in compensated magnets.