Single crystal high frequency EPR spectroscopy has been employed on a truly axial Single Molecule Magnet of formula [Mn12O12(tBu-CH2CO2)16(CH3OH)4].2CH3OH to investigate the origin of the transverse magnetic anisotropy, a crucial parameter that rules the quantum tunneling of the magnetization. The crystal structure, including the absolute structure of the crystal used for EPR experiments, has been fully determined and found to belong to I-4 tetragonal space group. The angular dependence of the resonance fields in the crystallographic ab plane shows the presence of high order tetragonal anisotropy and strong Ms dependence with the second-highest-field transition being angular independent. This was rationalized including competing fourth- and sixth-order transverse parameters in a Giant Spin Hamiltonian which describes the magnetic anisotropy in the ground S=10 spin state of the cluster. To establish the origin of these anisotropy terms the experimental results have been further analyzed using a simplified Multi Spin Hamiltonian which takes into account the exchange interactions and the single ion magnetic anisotropy of the MnIII centers. It has been possible to establish magneto-structural correlations with Spin Hamiltonian parameters up to the sixth order. Transverse anisotropy in axial SMMs was found to originate from the multi-spin nature of the system and from the break-down of the strong exchange approximation. The tilting of the single-ion easy axes of magnetization with respect to the four-fold molecular axis of the cluster plays the major role in determining the transverse anisotropy. Counter-intuitively the projections of the single ion easy axes on the ab plane correspond to hard axes of magnetization.