The advection of the interface, the mass density as well as the momentum density play a central role in any two-phase flow model. In this work we present and analyse a dimensionally unsplit advection method for the interface, mass and (centred) momentum density in the presence of a sharp interface that is based on the geometric VOF method, following the ideas in [Rudman 1998]. Although mass and momentum advection methods based on the geometric VOF method are used in practise, a thorough analysis of their structure preserving properties is lacking. Besides presenting such an analysis in this work, our proposed method is sufficiently general to be applied to arbitrary velocity fields and general polytopal meshes, while maintaining their desirable properties. In particular, we show that the presented method yields a conserved phase volume if the velocity field is divergence-free, while the volume fraction is guaranteed to remain bounded between zero and one. Mass and momentum are conserved exactly per phase, and the mass densities are guaranteed to remain positive. Moreover, the momentum density is advected in a minimally dissipative manner, resulting in exact conservation of the kinetic energy in the semi-discrete limit. The method can be applied to a general polytopal tessellation of the domain and does not assume a divergence-free velocity field. Numerical results of passive scalar advection are presented which confirm the stated claims.