Two digit cubes that show every date of a month

You have two blank cubes and may write one digit on each of their twelve faces. Placed side by side, the cubes must be able to display every day of a month as a two-digit number, from $01$ up to $31$. How should the faces be labelled?

Cube one: $0,1,2,3,4,5$; cube two: $0,1,2,6,7,8$, with the $6$ turned upside down to serve as $9$. Dates $11$ and $22$ force $1$ and $2$ onto both cubes, and $01$–$09$ force $0$ onto both as well, leaving only six free faces for the seven digits $3$–$9$. The trick is that one face can read as either $6$ or $9$.