  The results of a Fourier analysis of high-precision Kepler photometry of 75
double-mode RR~Lyrae (RRd) stars observed during NASA's K2 Mission (2014-18)
are presented. Seventy-two of the stars are `classical' RRd (cRRd) stars lying
along a well-defined curve in the Petersen diagram and showing no evidence of
Blazhko modulations. The remaining three stars are `anomalous' RRd (aRRd) stars
that lie well below the cRRd curve in the Petersen diagram. These stars have
larger fundamental-mode amplitudes than first-overtone amplitudes and exhibit
Blazhko variations. Period-amplitude relations for the individual pulsation
components of the cRRd stars are examined, as well as correlations involving
Fourier phase-difference and amplitude-ratio parameters that characterize the
light curves for the two radial modes. A simple statistical model relating the
fundamental (P0) and first-overtone (P1) periods to [Fe/H] provides insight
into the functional form of the Petersen diagram. A calibration equation for
estimating [Fe/H]phot abundances of `classical' RRd stars is derived by
inverting the model and using 211 field and 57 globular cluster cRRd stars with
spectroscopic metallicities to estimate the model coefficients. The equation is
used to obtain [Fe/H]phot for the full sample of 72 K2 cRRd stars and for 2130
cRRd stars observed by the ESA Gaia Mission. Of the 49 K2 cRRd stars that are
in the Gaia DR3 catalogue only five were found to be correctly classified, the
remainder having been misclassified `RRc' or `RRab'.
