Insurance carriers are the poster child for counter-cyclical business (https://en.wikipedia.org/wiki/Insurance_cycle). (Tangentially, article is interestingly pretty opinionated for wikipedia! Strange.)

Staple goods and foods are also generally counter-cyclical, because in times of prosperity, people tend to eat out and travel more.

There aren't many examples of VC-backed companies in this category, because it's very very hard for upstarts to break into counter-cyclical industries.

Gaming and other cheap (fixed-price) entertainment is often counter-cyclical. When people get laid off they have more time to fill; similarly, when the general mood is depressed, people look for things that will distract them from their miserable predicament.

There're some other niche examples, eg. liquidation specialists and bankruptcy lawyers. These are pretty obvious ones: anything that people do when they lack money tends to get a boost in a recession.

Government is a common counter-cyclical.