For continuous covariates it is a common to fit them as linear terms. However, one sometime finds that they are first used to define categories. For example, various age groups might be defined using age. If these are fitted then it is usual to fit them as a factor. The levels of the factor are then implicitly taken as unordered.
Similarly, for directly defined ordered covariates it is common to fit the covariate as an unordered factor.
A more rational approach might be to define orthogonal contrasts on the covariate, gradually fitting higher terms. If the covariate was continuous, then another possibility, less extreme than simply using a liner predictor would be to fit a spline.
In this project various strategies for fitting ordered categorical covariates will be fitted with a view to investigating possible biases and gains or losses in efficiencies.