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We study composites of cellulose nanocrystals (CNCs) in an ionomer matrix of poly(ethylene-stat-sodium acrylate) and find that direct cellulose/cellulose interactions in the composite are not a requirement for achieving reinforcement. While isotropic composites only show a slightly enhanced stiffness compared to the neat ionomer, a more substantial increase in Young's modulus by a factor of up to 5 is achieved by uniaxial alignment of the composites through melt spinning. The orientation of CNC in melt-spun composites reduces the probability of cellulose/cellulose interactions, which suggests that cellulose/polymer interactions must be present that lead to the observed reinforcement. Molecular dynamics simulations confirm strong cellulose/polymer interactions in the form of ionic interactions as well as hydrogen bonding. These cellulose/polymer interactions facilitate efficient stress transfer, leading to the high reinforcing effect of CNC, while cellulose/cellulose interactions play a minor role in the mechanical response of the composite.