I found evidence that the X-ray surface flux of intermediate-mass G and K giants is correlated with their rotation period and Rossby number. Confidence in the degree of correlation is significantly higher when stellar gravity is taken into account. An empirical relation is found that accounts for the X-ray luminosity evolution of single intermediate-mass giants and giants in close or long-period binary systems, such as RS CVn-type systems, as they evolve off the main sequence towards the top of red giant branch.