We have created oriented polypropylene materials by die-drawing to draw ratios up to ~16, increasing tensile stiffness and strength as measured along the direction of drawing by factors up to 12. We have performed nanoindentation testing on these materials by indenting in all 3 directions; both along the drawing axis (axial) and normal to it (transverse). These measurements provided values of axial indentation modulus that exceeded those for transverse measurement by up to 60%. To analyse the indentation experiments in terms of macroscopic properties, the anisotropy of the elastic properties must be taken into account. Also, the modulus of elasticity as measured along the draw direction depends on whether the measurement is in tension or compression; tensile and compressive moduli differ by a factor of ~5 at the highest draw ratio. A material model that accounts for both these effects, including both orthotropic elasticity and asymmetry with respect to tension and compression. This model has been incorporated into Finite Element simulations to create a unique analysis of nanoindentation of an oriented polymer. Elastic properties of our oriented polypropylenes were obtained by a combination of static testing (tensile and compressive) and published ultrasonic data. The resulting elastic constants were used as input for the nanoindentation simulations with a Berkovich indenter to give successful predictions of both axial and transverse measurements of indentation modulus. The significance of the tension/compression asymmetry was demonstrated by comparing these predictions with those obtained using tensile data only, which gave predictions of indentation modulus higher by up to 70%.