Minimal-order inverses of a linear continuous time-invariant system are defined and an algebraic algorithm for constructing them is presented. A necessary and sufficient condition for the applicability of the algorithm is shown to be the same as that for the invertibility of the given system. Some important properties of the minimal-order inverses relevant to the design of linear multivariable systems are derived. In particular the characteristic polynomials of all minimal-order inverses of a given system are shown to be identical and invariant under duality, coordinate transformations and proportional state feedback.