We present a broad range of measurements of the angular orientation $\theta_0(t)$ of the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection as a function of time. We used two cylindrical samples of different overall sizes, but each with its diameter nearly equal to its height. The fluid was water with a Prandtl number of 4.38. The time series $\theta_0(t)$ consisted of meanderings similar to a diffusive process, but in addition contained large and irregular spontaneous reorientation events through angles $\uDelta \theta$. We found that reorientations can occur by two distinct mechanisms. One consists of a rotation of the circulation plane without any major reduction of the circulation strength. The other involves a cessation of the circulation, followed by a restart in a randomly chosen new direction. Rotations occurred an order of magnitude more frequently than cessations. Rotations occurred with a monotonically decreasing probability distribution $p(\uDelta \theta)$, i.e. there was no dominant value of $\uDelta \theta$ and small $\uDelta \theta$ were more common than large ones. For cessations, $p(\uDelta\theta)$ was uniform, suggesting that information of $\theta_0(t)$ is lost during cessations. Both rotations and cessations have Poissonian statistics in time, and can occur at any $\theta_0$. The average azimuthal rotation rate $|\skew4\dot\theta|$ increased as the circulation strength of the LSC decreased. Tilting the sample relative to gravity significantly reduced the frequency of occurrence of both rotations and cessations.