The probability distribution of the winding angle theta of a planar self-avoiding walk has been known exactly for a long time: it has a Gaussian shape with a variance growing as <theta(2)> similar to ln L. For the three-dimensional case of a walk winding around a bar, the same scaling is suggested, based on a first-order epsilon-expansion. We tested this three-dimensional case by means of Monte Carlo simulations up to length L approximate to 25 000 and using exact enumeration data for sizes L <= 20. We find that the variance of the winding angle scales as <theta(2)> (ln L)(2 alpha), with alpha = 0.75(1). The ratio gamma = <theta(4)>/<theta(2)>(2) = 3.74(5) is incompatible with the Gaussian value gamma = 3, but consistent with the observation that the tail of the probability distribution function p(theta) is found to decrease more slowly than a Gaussian function. These findings are at odds with the existing first-order epsilon-expansion results.