Deleting ZenCache and reinstalling the latest Pro version (rather than upgrading from a previous version) appears to have resolved the issue.
@raamdev, downgrading to v150409 fixed the problem. However, that’s the free version so I’ve also lost all the Pro features. Is it still possible to access old versions of ZenCache Pro?
That’s nice, but every version prior to the latest has worked fine with IIS, going back a year-plus. No change in server configuration. Seems likely enough to be a bug introduced in a recent change to the plugin or WordPress to be worth investigating. “Don’t use Windows” noted.