  In this work we provide a framework that connects the co-rotating and counter
rotating $f$-mode frequencies of rotating neutron stars with their stellar
structure. The accurate computation of these modes for realistic equations of
state has been presented recently and they are here used as input for a
Bayesian analysis of the inverse problem. This allows to quantitatively
reconstruct basic neutron star parameters, such as the mass, radius, rotation
rate or universal scaling parameters. We find that future observations of both
$f$-mode frequencies, in combination with a Bayesian analysis, would provide a
promising direction to solve the inverse stellar problem. We provide two
complementary approaches, one that is equation of state dependent and one that
only uses universal scaling relations. We discuss advantages and disadvantages
of each approach, such as possible bias and robustness. The focus is on
astrophysically motivated scenarios in which informed prior information on the
neutron star mass or rotation rate can be provided and study how they impact
the results.
