The orientation relations (ORs) of the cubic icosahedral quasicrystal approximant phase α-Al(Fe,Mn)Si have been studied after low temperature annealing of a 3xxx wrought aluminium alloy by transmission electron microscopy. From diffraction studies it was verified that the most commonly observed OR for the α-Al(Fe,Mn)Si dispersoids is [1\bar 11]α // [1\bar 11]Al, (5\bar 2\bar 7)α // (011)Al. This orientation could be explained by assuming that the internal Mackay icosahedron (MI) in the α-phase has a fixed orientation in relation to Al, similar to that of the icosahedral quasi-crystals existing in this alloy system. It is shown that mirroring of the normal-to-high-symmetry icosahedral directions of the MI explains the alternative orientations, which are therefore likely to be caused by twinning of the fixed MI. Only one exception was found, which was related to the Bergman icosahedron internal to the T-phase of the Al-Mg-Zn system.