The adsorption of formic acid (HCOOH) on the rutile TiO2(110) surface, where HCOOH dissociates into formate (HCOO) and hydrogen (H), has been analyzed with first-principles density functional calculations. A p(2 x 1) geometry can be observed for high exposure, while only local c(4 x 2) domains are detected for lower exposure. After desorption of some HCOOH at higher temperatures the p(2 x 1) geometry becomes unstable and the remaining molecules form a disordered structure. A direct and simultaneous observation of PI and HCOO is difficult, but the calculations show that coadsorption of H is necessary to stabilize the p(2 x 1) geometry. Some of the vibrational modes also show a slight dependence on the supply of H and might give experimentally accessible hints concerning the role of H. An interesting aspect concerns the surface diffusion of HCOO: while HCOO molecules behave rather immobile on the surface in the c(4 x 2) domains and in the disordered structure, they show a high mobility along the [001] direction in a p(3 x 1)-like environment. As it turns out, the reason for these findings can be found in the complex movement of HCOO molecules, which includes the coadsorbed HCOO-H compound as a whole.