A new source mechanism is proposed for the 'reflected' ion beams observed in the foreshock region of the earth's bow shock. In our model the beams originate in the magnetosheath downstream of the qausi-perpendicular portion of the shock. The quasi-perpendicular shock transition is characterized by two downstream ion populations including high-energy gyrating ions in addition to the directly transmitted anisotropic ions. We show by particle simulations that this highly anisotropic downstream ion distribution (T⊥/T∥ ≫1) can excite electromagnetic ion cyclotron waves which, in turn, pitch angle scatter the gyrating ions in a few ion gyroperiods. As a result, some ions acquire large parallel velocities and move fast enough along the convecting downstream magnetic field to escape back across the bow shock into the upstream region. The distribution of escaping ions calculated by using the pitch-angle-scattered ions, as a source, becomes a beam with a large temperature anisotropy T⊥ ~3--5 T∥ and a mean velocity along the magnetic field of about twice that of the solar wind velocity. A significant result is the presence of the maximum angle &thgr;nB =&thgr;c above which no ions can escape, where &thgr;nB is the angle between the shock normal and the interplanetary magnetic field. A wide peak of constant escaping ion flux is formed below &thgr;c whose number density is 1--2% of that of the solar wind. These results are in general agreement with the ISEE observations of the 'reflected' ions.