  It is found that $15$ different types of two-qubit $X$-states split naturally
into two sets (of cardinality $9$ and $6$) once their entanglement properties
are taken into account. We {characterize both the validity and entangled nature
of the $X$-states with maximally-mixed subsystems in terms of certain
parameters} and show that their properties are related to a special class of
geometric hyperplanes of the symplectic polar space of order two and rank two.
Finally, we introduce the concept of hyperplane-states and briefly address
their non-local properties.
