  The translational motion of anisotropic and self-propelled colloidal
particles is closely linked with the particle's orientation and its rotational
Brownian motion. In the overdamped limit, the stochastic evolution of the
orientation vector follows a diffusion process on the unit sphere and is
characterised by an orientation-dependent ("multiplicative") noise. As a
consequence, the corresponding Langevin equation attains different forms
depending on whether It\=o's or Stratonovich's stochastic calculus is used. We
clarify that both forms are equivalent and derive them from a geometric
construction of Brownian motion on the unit sphere, based on infinitesimal
random rotations. Our approach suggests further a geometric integration scheme
for rotational Brownian motion, which preserves the normalisation constraint of
the orientation vector exactly. We show that a simple implementation of the
scheme converges weakly at order 1 of the integration time step, and we outline
an advanced variant of the scheme that is weakly exact for an arbitrarily large
time step. The discussion is restricted to time-homogeneous rotational Brownian
motion (i.e., constant rotational diffusion tensor), which is relevant for
chemically anisotropic spheres such as self-propelled Janus particles.
