When a rigid right-angled lever experiences equal and opposite torques as measured by an observer *O*′ at rest with respect to the lever in frame *S*′, the usual transformation of these torques to another Lorentz frame *S* makes it appear that the torques are not balanced for an observer *0* at rest in *S*. This paradox was first pointed out by Lewis and Tolman. The consensus among textbook authors is that a net torque does exist on the lever according to the general Lorentz observer. The present paper is a basic reexamination of both the Lewis-Tolman lever paradox and another, closely related, paradox: equal and opposite torques applied at the same point on a rigid square. The basic approach herein is to construct for observer *O* a four-tensor $I\mu \nu $ to represent the relative four-torque about a point *P*′ fixed in frame *S*′ which has a constant speed $\nu \u2009=\u2009\beta c$ along the $+x$ axis of *S*. For some unknown reason, this tensor seems never to have been defined before. The space part of $I\mu \nu $ reduces to the corresponding classical definition of torque in three-space if $\nu \u2009=\u20090$. Based on $I\mu \nu $ a comparison of the torque components for two different Lorentz observers shows that *no* torque exists on the square or on the lever for *any* Lorentz observer if no net torque exists for the observer at rest with respect to the square or lever. This analysis suggests a new (covariant) definition for the torque $I$ about a point $P\u2032a$ in three-space: $I\u2009=\u2009l\u2009\xd7\u2009f$, where $l$ is the space part of the four-vector $l\mu \u2009=\u2009(\gamma 2lx,\u2009ly,\u2009lz,\u2009i\gamma 2\beta lx)$ and where $f$ is the space part of the four-vector $f\mu \u2009=\u2009(\gamma F,\u2009i\gamma \beta Fx)$. Here, a force $F$ is applied at a point $P\u2032b$ (stationary in *S*′) whose displacement from $P\u2032a$ (also stationary in *S*′) is 1. Both the force $F$ and the displacement 1 are as measured by observer *O* at rest in *S*; and $I$ is the torque about $P\u2032a$ according to *O*.

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