Bilinguals with a high proficiency in their first (L1) and second language (L2) often show comparable reaction times when switching from their L1 to L2 and vice-versa (“symmetrical switch costs”). However, the neurophysiological signatures supporting this effect are not well understood. Here, we ran two separate experiments and assessed behavioral and MEG responses in highly proficient Spanish-Basque bilinguals while they overtly name pictures in a mixed-language context. In the behavioral experiment, bilinguals were slower when naming items in switch relative to non-switch trials, and this switch cost was comparable for both languages (symmetrical). The MEG experiment mimicked the behavioral one, with switch trials showing more desynchronization than non-switch trials across languages (symmetric neural cost) in the alpha band (8–13 Hz). Source-localization revealed the engagement of right parietal and premotor areas, which have been linked to language selection and inhibitory control; and of the left anterior temporal lobe (ATL), a cross-linguistic region housing conceptual knowledge that generalizes across languages. Our results suggest that highly proficient bilinguals implement a language-independent mechanism, supported by alpha oscillations, which is involved in cue-based language selection and facilitates conceptually-driven lexical access in the ATL, possibly by inhibiting non-target lexical items or disinhibiting target ones.