Orthogonal regression $M$-estimates are considered from a bias robust point of view. Their maximum bias over epsilon-contamination neighborhoods is characterized, and maximum bias curves are computed. The most bias robust orthogonal regression $M$-estimate is derived and shown to be a "mode type" estimate; for instance, in the two-dimensional case this estimate can be computed by locating a strip of fixed width covering the maximum number of data points. It will be shown that, although orthogonal regression $M$-estimates with bounded loss function have unbounded influence function, the derivative of their maximum bias curve at zero is finite. Finally, an implicit formula for an upper bound for the breakdown point of all orthogonal regression $M$-estimates is found. The upper bound, which depends on the signal-to-noise ratio, is sharp and attained by the most bias robust estimate.