I typed this out in reply to your private message before reading the comments, but it’s similar to some of the ideas above.
After memorizing the numbers in a 37-location memory palace or peg list, you could add location numbers: How to add position number?
If the current location number is even, it’s red. If it’s odd, it’s black (with the exception of position 1).
One way to get the location between the two opposite numbers:
- If the current location number is in the range 1-18, then add 18.5.
- If the current location number is in the range 19-37, then subtract 18.5.
Example 1: If you’re at position 12 (number value: 27), add 18.5 to get 30.5. That’s the line between the two target locations. Hop over to location #30 and look at the values in locations #30 (even, so it’s red) and #31 (odd, so it’s black). The numbers are 18 and 29.
Example 2: If you’re at position 34 (number value: 12), subtract 18.5 to get 15.5. Then hop over to location #15 and look at the values in locations #15 (odd, so it’s black) and #16 (even, so it’s red). The results there are 11 and 30.
Just some ideas. I didn’t test it with locations. 
Edit: it might be interesting to create a circular 37-location memory palace that generally matches the layout of the wheel. Then it might even be easier to mentally hop across the room.