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you can be right-handed and play 1B, but you better hit, because the 3-6-x DP and the 3B charging on a bunt barehand throw (keeping the throw clear of the runner) are easier for the LH, amongst other plays.
That's why I think Montero will never be anything but a back-up 1B - he lacks the athleticism to compensate for having the glove on the runner's (baseline) side. But if he hits, a DH/back-up C/back-up 1B is not a bad thing (see Martinez, Victor, or Napoli, Mike at your local B-R.com).