When would you report the median instead of the mean?

Report the median when the distribution is skewed or has outliers, which covers most business metrics: revenue per customer, session duration, page load time, salary. The mean is dragged by a handful of extreme values, so it can describe almost nobody. The median describes the typical case. Where the total matters, such as revenue forecasting, the mean is still the right input because it scales to the sum.

Whenever the distribution is skewed, which in practice is most things I work with. Revenue per customer, time on site, order value, they all have a long right tail, so the mean sits somewhere above what a typical customer actually does. We had an average order value that jumped 30% one month and it turned out to be two wholesale orders in a consumer dataset. The median did not move at all, which was the honest picture. That said, I would not say median is always better. If finance is forecasting total revenue, the mean is exactly what they need, because mean times count gives the total and the median does not. So I pick based on the question: typical experience means median, totals and budgets mean mean. What I actually prefer to show is the distribution, or at least p25, p50 and p90 side by side, because a single number for a skewed metric always loses the thing the audience needs to know.