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The highly elongated necks, and often tails, of sauropod dinosaurs were composed of concavo-convex vertebrae that provided stability without compromising mobility. Polarities of these concavo-convex joints in the neck and tail are anatomically opposite one another but mechanically equivalent. Opisthocoelous cervical vertebrae and procoelous caudal vertebrae have the convex articular face directed away from the body and the concave articular face directed toward the body. This “sauropod-type” polarity is hypothesized to be (1) more resistant to fracturing of the cotylar rim and (2) better stabilized against joint failure by rotation than the opposite polarity. We used physical models to test these two functional hypotheses. Photoelastic analysis of model centra loaded as cantilevers reveals that neither polarity better resists fracture of the cotylar rim; strain magnitude and localization are similar in both polarities. We assessed the rotational stability of concavo-convex joints using pairs of concavo-convex centra loaded near the joint. Sauropod-type joints withstood significantly greater weight before failure occurred, a pattern we interpret to be dependent on the position of the center of rotation, which is always within the convex part of the concavo-convex joint. In sauropod-type joints, the free centrum rotates about a center of rotation that lies within the more stable proximal centrum. In contrast, the opposite polarity results in a free centrum that rotates about an internal point; when the condyle rotates down and out of joint, the distal end rotates back toward the body, unopposed by ligamentous support. Sauropod-type joints remained stable with greater mobility, more mechanically advantageous tensile element insertions, and greater distal loads than the opposite polarity. The advantages conferred by this joint polarity would have facilitated the evolution of hyperelongated necks and tails by sauropods. Polarity of concavo-convex joints of the appendicular skeleton (e.g., hip, shoulder) is also consistent with the demands of rotational stability.