  Tidal interactions play an important role in many astrophysical systems, but
uncertainties regarding the tides of rapidly rotating, centrifugally distorted
stars and gaseous planets remain. We have developed a precise method for
computing the dynamical, non-dissipative tidal response of rotating planets and
stars, based on summation over contributions from normal modes driven by the
tidal potential. We calculate the normal modes of isentropic polytropes
rotating at up to $\simeq90\%$ of their critical breakup rotation rates, and
tabulate fits to mode frequencies and tidal overlap coefficients that can be
used to compute the frequency-dependent, non-dissipative tidal response (via
potential Love numbers $k_{\ell m}$). Although fundamental modes (f-modes)
possess dominant tidal overlap coefficients at (nearly) all rotation rates, we
find that the strong coupling of retrograde inertial modes (i-modes) to
tesseral ($\ell>|m|$) components of the tidal potential produces resonances
that may be relevant to gas giants like Jupiter and Saturn. The coupling of
f-modes in rapid rotators to multiple components of both the driving tidal
potential and the induced gravitational field also affect the tesseral
response, leading to significant deviations from treatments of rotation that
neglect centrifugal distortion and high-order corrections. For very rapid
rotation rates ($\gtrsim 70\%$ of breakup), mixing between prograde f-modes and
i-modes significantly enhances the sectoral ($\ell=|m|$) tidal overlap of the
latter. The tidal response of very rapidly rotating, centrifugally distorted
planets or stars can also be modified by resonant sectoral f-modes that are
secularly unstable via the Chandrasekhar-Friedman-Schutz (CFS) mechanism.
