Telerobotic systems enable humans to explore and manipulate remote environments for applications such as surgery and disaster response, but few such systems provide the operator with cutaneous feedback. This article presents a novel approach to remote cutaneous interaction; our method is compatible with any fingertip tactile sensor and any mechanical tactile display device, and it does not require a position/force or skin deformation model. Instead, it directly maps the sensed stimuli to the best possible input commands for the device's motors using a data set recorded with the tactile sensor inside the device. As a proof of concept, we considered a haptic system composed of a BioTac tactile sensor, in charge of measuring contact deformations, and a custom 3-DoF cutaneous device with a flat contact platform, in charge of applying deformations to the user's fingertip. To validate the proposed approach and discover its inherent tradeoffs, we carried out two remote tactile interaction experiments. The first one evaluated the error between the tactile sensations registered by the BioTac in a remote environment and the sensations created by the cutaneous device for six representative tactile interactions and 27 variations of the display algorithm. The normalized average errors in the best condition were 3.0 percent of the BioTac's full 12-bit scale. The second experiment evaluated human subjects' experiences for the same six remote interactions and eight algorithm variations. The average subjective rating for the best algorithm variation was 8.2 out of 10, where 10 is best.