Aims. We model the effects of the spiral arms of the Milky Way on the disk stellar kinematics in the Gaia observable space. We also estimate the Gaia capabilities of detecting the predicted signatures. Methods. We use both controlled orbital integrations in analytic potentials and self-consistent simulations. We introduce a new strategy to investigate the effects of spiral arms, which consists of comparing the stellar kinematics of symmetric Galactic longitudes (+l and −l), in particular the median transverse velocity as determined from parallaxes and proper motions. This approach does not require the assumption of an axisymmetric model because it involves an internal comparison of the data. Results. The typical differences between the transverse velocity in symmetric longitudes in the models are of the order of ~2 km s-1, but can be larger than 10 km s-1 for certain longitudes and distances. The longitudes close to the Galactic centre and to the anti-centre are those with larger and smaller differences, respectively. The differences between the kinematics for +l and −l show clear trends that depend strongly on the properties of spiral arms. Thus, this method can be used to quantify the importance of the effects of spiral arms on the orbits of stars in the different regions of the disk, and to constrain the location of the arms, main resonances and, thus, pattern speed. Moreover, the method allows us to test different origin scenarios of spiral arms and the dynamical nature of the spiral structure (e.g. grand design versus transient multiple arms). We estimate the number of stars of each spectral type that Gaia will observe in certain representative Galactic longitudes, their characteristic errors in distance and transverse velocity, and the error in computing the median velocity as a function of distance. We will be able to measure the median transverse velocity exclusively with Gaia data, with precision smaller than ~1 km s-1 up to distances of ~4-6 kpc for certain giant stars, and up to ~2-4 kpc and better kinematic precision (≲0.5 km s-1) for certain sub-giants and dwarfs. These are enough to measure the typical signatures seen in the models. Conclusions. The Gaia catalogue will allow us to use the presented approach successfully and improve significantly upon current studies of the dynamics of the spiral arms of our Galaxy. We also show that a similar strategy can be used with line-of-sight velocities, which could be applied to Gaia data and to upcoming spectroscopic surveys.