We present new deep, high-resolution radio images of the diffuse minihalo in the cool core of the galaxy cluster RX J1720.1+2638. The images have been obtained with the Giant Metrewave Radio Telescope at 317, 617 and 1280 MHz and with the Very Large Array at 1.5, 4.9 and 8.4 GHz, with angular resolutions ranging from 1 to 10. This represents the best radio spectral and imaging dataset for any minihalo. Most of the radio flux of the minihalo arises from a bright central component with a maximum radius of ~80 kpc. A fainter tail of emission extends out from the central component to form a spiral-shaped structure with a length of ~230 kpc, seen at frequencies 1.5 GHz and below. We find indication of a possible steepening of the total radio spectrum of the minihalo at high frequencies. Furthermore, a spectral index image shows that the spectrum of the diffuse emission steepens with the increasing distance along the tail. A striking spatial correlation is observed between the minihalo emission and two cold fronts visible in the Chandra X-ray image of this cool core. These cold fronts confine the minihalo, as also seen in numerical simulations of minihalo formation by sloshing-induced turbulence. All these observations favor the hypothesis that the radio emitting electrons in cluster cool cores are produced by turbulent reacceleration.