Mark, are you sure you are not missing out one of the purl rows? I can't see any other way that your pattern would be 'flipping'. The right side of the mitten will have your long tail from casting on at the start of the row and the stitches will be v-shaped for the most part, apart from the occasional bump which would be one of the purl stitches. When you are going to be doing mostly knit stitches in a row, make sure the side facing you has v-shaped stitches and when you are going to be purling all across the row, make sure the side facing you has the purl bumps.
Hope that helps. Have you taught your wife the basic knit stitch? She could try out a basic sample square or any other shape she cares to choose.