A novel set of 5 strain invariants for materials exhibiting transversely isotropic behavior with respect to a reference configuration is developed through analysis of physical attributes of deformation. Experimental advantage for hyperelastic materials is demonstrated by showing that common tests can directly determine terms in W, the strain energy per unit reference volume. An analysis of symmetry allows the general form of W to be refined a priori. Moreover, this kinematics framework is potentially useful for solving inverse problems since the 5 response terms in the Cauchy stress are mostly orthogonal (9 of the 10 mutual inner products vanish). For small deformation they are fully orthogonal (all 10 inner products vanish). A response term in consists of an invariant response function multiplied by its associated kinematic tensor.