We study the Langevin dynamics of the spherical Sherrington--Kirkpatrick model, the diffusion on the sphere of radius $\sqrt N$ in $\mathbb R^N$ that is reversible with respect to the Gibbs measure with density proportional to $\exp(\beta x^{\mathsf T}Wx/\sqrt2)$, where $W$ is an $N\times N$ matrix from the Gaussian orthogonal ensemble. The dynamics is started from the uniform distribution on the sphere, independently of $W$. We determine the asymptotic quenched total-variation distance to equilibrium as a function of time in the fixed-temperature, mesoscopic, critical-window, and infinite-temperature crossover regimes described below, with convergence in probability or in distribution with respect to $W$. Let $\beta_c=1/\sqrt2$ be the critical inverse temperature. For $\beta<\beta_c$ there is cutoff with a Gaussian profile at fixed $\beta\in(0,\beta_c)$, along every mesoscopic approach of $\beta$ to $\beta_c$ from below, and in the infinite-temperature crossover when $N\beta_N^2\to\infty$; when $N\beta_N^2$ stays bounded, the distance relaxes over bounded times and there is no cutoff. At fixed $\beta\in(0,\beta_c)$ the mixing time is $(\log N-(3/2)\log\log N+O(1))/(4(1-\sqrt2\beta)^2)$, and the correction $-(3/2)\log\log N$ comes from the edge of the spectrum of $W$. For $\beta>\beta_c$, at fixed $\beta$ and along every mesoscopic approach to $\beta_c$ from above, there is cutoff at a time set by the gap between the two largest eigenvalues of $W$, of order $N^{2/3}\log N$ at fixed $\beta$. The profile is the total-variation distance between a standard Gaussian variable and its sum with an independent Cauchy variable; the Cauchy variable is the ratio of the first two eigencoordinates at an early stage of the evolution. In the critical window $\sqrt2\beta=1+\sigma N^{-1/3}$, the distance at time $N^{2/3}s$ converges in distribution to a random profile determined by the Airy point process, and there is no cutoff. As $\sigma\to\pm\infty$, this profile, suitably recentered and rescaled, converges to the profiles of the two mesoscopic regimes.