A detailed investigation of the phase diagram of the mixtures of the two ionic liquids N-trimethyl-N-propylammonium bis(trifluoromethylsulfonyl)imide (TMPA-TFSI) and N-trimethyl-N-hexylammonium bis(trifluoromethylsulfonyl)imide (TMHA-TFSI) has been performed in the temperature range between 140 and 330 K by means of DSC and infrared spectroscopy measurements. In the low temperature crystalline states, a large concentration of the trans-TFSI conformer is present in TMPA-TFSI, while almost only cis-TFSI is retained in TMHA-TFSI. For the mixtures (TMPA-TFSI)100-x (TMHA-TFSI)x, at concentrations close to the extremes, solid crystalline phases are still present and they show a strong predominance of the trans conformer of the TFSI ion for x < 15 or a large concentration of the cis conformer of TFSI for x > 85. At intermediate concentrations (33 < x < 67) no crystalline phase develops at low temperatures and both conformers of TFSI survive in the whole temperature range investigated here. We suggest that the competition between the two TFSI conformers at low temperatures can be the origin of the lack of crystalline phases for intermediate concentrations and can be exploited as a valid tool to tailor the physical properties of the mixtures of ionic liquids.