The first measurements of the scattering parameters of K pairs in all three charge combinations (K+, K−, and K0S) are presented. The results are achieved through a femtoscopic analysis of K correlations in Pb-Pb collisions at √sNN = 2.76 TeV recorded by ALICE at the Large Hadron Collider. The femtoscopic correlations result from strong final-state interactions and are fit with a parametrization allowing for both the characterization of the pair emission source and the measurement of the scattering parameters for the particle pairs. Extensive studies with the THERMINATOR 2 event generator provide a good description of the nonfemtoscopic background, which results mainly from collective effects, with unprecedented precision. Furthermore, together with HIJING simulations, this model is used to account for contributions from residual correlations induced by feed-down from particle decays. The extracted scattering parameters indicate that the strong force is repulsive in the K+ interaction and attractive in the K− interaction. The data hint that the K0S interaction is attractive ; however, the uncertainty of the result does not permit such a decisive conclusion. The results suggest an effect arising either from different quark-antiquark interactions between the pairs (ss in K+ and uu in K−) or from different net strangeness for each system (S = 0 for K+, and S = −2 for K−). Finally, the K systems exhibit source radii larger than expected from extrapolation from identical particl femtoscopic studies. This effect is interpreted as resulting from the separation in space-time of the single-particle and K source distributions.