We present a lattice calculation of the leading-order electromagnetic and strong isospin-breaking corrections to the hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon. We employ the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with $N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing ($a \simeq 0.062, 0.082, 0.089$ fm) with pion masses between $\simeq 210$ and $\simeq 450$ MeV. The results are obtained adopting the RM123 approach in the quenched-QED approximation, which neglects the charges of the sea quarks. Quark disconnected diagrams are not included. After the extrapolations to the physical pion mass and to the continuum and infinite-volume limits the contributions of the light, strange and charm quarks are respectively equal to $��a_��^{\rm HVP}(ud) = 7.1 ~ (2.5) \cdot 10^{-10}$, $��a_��^{\rm HVP}(s) = -0.0053 ~ (33) \cdot 10^{-10}$ and $��a_��^{\rm HVP}(c) = 0.0182 ~ (36) \cdot 10^{-10}$. At leading order in $��_{em}$ and $(m_d - m_u) / ��_{QCD}$ we obtain $��a_��^{\rm HVP}(udsc) = 7.1 ~ (2.9) \cdot 10^{-10}$, which is currently the most accurate determination of the isospin-breaking corrections to $a_��^{\rm HVP}$.

23 pages, 7 figures, 5 tables. Version to appear in PRD. A bug in the update of the strange and charm contributions is removed and an extended discussion on the identification of the ground-state is included. arXiv admin note: text overlap with arXiv:1808.00887, arXiv:1707.03019