Datasets of mobile phone trajectories collected by network operators offer an unprecedented opportunity to discover new knowledge from the activity of large populations of millions. However, publishing such trajectories also raises significant privacy concerns, as they contain personal data in the form of individual movement patterns. Privacy risks induce network operators to enforce restrictive confidential agreements in the rare occasions when they grant access to collected trajectories, whereas a less involved circulation of these data would fuel research and enable reproducibility in many disciplines. In this work, we contribute a building block towards the design of privacy-preserving datasets of mobile phone trajectories that are truthful at the record level. We present GLOVE, an algorithm that implements k-anonymity, hence solving the crucial unicity problem that affects this type of data while ensuring that the anonymized trajectories correspond to real-life users. GLOVE builds on original insights about the root causes behind the undesirable unicity of mobile phone trajectories, and leverages generalization and suppression to remove them. Proof-of-concept validations with large-scale real-world datasets demonstrate that the approach adopted by GLOVE allows preserving a substantial level of accuracy in the data, higher than that granted by previous methodologies.