Using first principles calculations, we study the microscopic origin of ferroelectricity (FE) induced by magnetic order in the orthorhombic HoMnO3. We obtain the largest ferroelectric polarization observed in the whole class of improper magnetic ferroelectrics to date. We find that the two proposed mechanisms for FE in multiferroics, lattice- and electronic-based, are simultaneously active in this compound: a large portion of the ferroelectric polarization arises due to quantum-mechanical effects of electron orbital polarization, in addition to the conventional polar atomic displacements. An interesting mechanism for switching the multiferroic domains by an electric field via a 180 degree coherent rotation of Mn spins is also proposed.