Given a contrast matrix, try and interpret the intercept. Will usually be either the grand mean, the mean of a reference level (e.g. contr.treatment), the unweighted mean of multiple levels. Anything else would indicate custom weights that the user provided, hence they should know how to interpret it.
Examples
interpret_intercept(contr.treatment(2)) # mean(1)
#> [1] "mean(1)"
interpret_intercept(contr.SAS(2)) # mean(2)
#> [1] "mean(2)"
interpret_intercept(contr.sum(2)) # grand mean
#> [1] "grand mean"
# Here there are 3 levels but the intercept is either an unweighted
# mean of 2 levels or a weighted mean of 2 levels
unweighted_intercept <-
solve(t(matrix(c(.5, .5, 0, -1, 1, 0, -1, 0, 1), nrow = 3)))[, 2:3]
weighted_intercept <-
solve(t(matrix(c(.8, .2, 0, -1, 1, 0, -1, 0, 1), nrow = 3)))[, 2:3]
interpret_intercept(unweighted_intercept) # mean(1,2)
#> [1] "mean(1,2)"
interpret_intercept(weighted_intercept) # custom weights
#> [1] "custom weights"