In this paper, we outline the mathematical and physical foundation of the 1991 hypothesis by R. M. Santilli of IsoRedShift (IRS), namely, a frequency shift of light toward the red characterized by the loss of energy by light to a cold medium without any relative motion between the source, the medium and the observer; we outline the corresponding foundations of Santilli's IsoBlueShift (IBS), namely, a frequency shift of light toward the blue characterized by the acquisition of energy by light from a hot medium without relative motion; we show the compatibility of Santilli's IRS and IBS with the axioms of special relativity under their proper mathematical formulations; we review the original experimental confirmations of IRS by Santilli in 2010 for a blue laser light in a tube containing air at pressure; we review the experimental confirmations of both IRS and IBS by G. West and G. Amato in 2011; we review the confirmatory measurements by Santilli, West and Amato done in 2012 on the West coast of Florida on the IRS origin of the redness of the Sun in the transition from the Zenith to the horizon; we present, for the first time, additional confirmatory measurements of IRS of Sunlight from the Zenith to Sunset done by the authors at the island of Kos, Greece, on September 20, 2012; we present, also for the first time, additional confirmatory measurements of IRS of Sunlight from Sunrise to the Zenith done in Cocoa Beach, East Coast of Florida, on October 20, 2012; we review the compatibility of Santilli's 1991 IRS hypothesis with Zwicky's 1929 hypothesis of Tired Light and identify their difference in the process originating the redshift; we recall the fit of cosmological redshift done by P. LaViolette in 1986 with Zwicky's Tired Light hypothesis; we confirm Santilli's 2010 argument according to which the IRS origin of the redness of the Sun at the horizon without relative motion is visual evidence on the expected absence of the expansion of the universe and of related conjectures; and we present the dismissal of various objections against Santilli's IRS and its interpretation of the cosmological redshift. In essence, we agree with Santilli that cosmologists should follow Galileo's teaching by establishing cosmological models via experiments on Earth prior to their application to the universe.