How do I encode day of the week as a predictor?

I have predictors in the form of days of the week (Sun, Mon, Tues, Wed, Thu, Fri, Sat).

I can't use 1-of-C encoding because it does not reflect the adjacency property correctly. Example: Predicting Tuesday when it was actually Wednesday shouldn't incur as large a cost as predicting Saturday when it was actually Wednesday.

I can't use ordinal regression because that doesn't reflect the modulo nature of my categories. Example: Predicting Sunday when it was actually Saturday would incur a huge cost even though they're adjacent.

In order words my data has a relative ordering, but there's no absolute ordering.