Do the partial derivatives of the compensated demand have an interpretation? When obtaining the Marshallian demand from a utility maximization problem we have a classification of normal goods, inferior goods, Giffen-goods, etc.. These are related to the first derivative of the Marshallian demand function with respect to prices or income.

My question is whether a similar classification exists for compensated demand functions of the expenditure minimization problem or whether the (same) goods classification can be derived via the compensated demand.
To the question on whether the same classification can be achieved through the hicksian demand function: in general I believe the answer is no. To see this, relate the classifications to the Slutsky decomposition.

(Changes in) Hicksian demand is related to the substitution effect from the Slutsky decomposition. Any standard textbook should provide a reference to this.

There are two broad classifications:

1.Ordinary and Giffen goods: These refer to marshallian demand changes when prices change. This is what we call the Total Effect in the Slutsky 
decomposition.


Normal and inferior goods: These refer to marshallian demand changes when income changes. This is by definition the Income Effect in the Slutsky decomposition.


As you see, the substitution effect, which is represented by changes in hicksian demand, is not a part of any of these classifications you mentioned.