A possible way is by checking the height and width of the image in pixels. If the height is higher than the width then you can say it's portrait or else landscape. The BitmapImage class has no method for telling you this right away.
Something as simple as
if (bitmap.PixelHeight > bitmap.PixelWidth) {
// portrait
} else {
// landscape
}
There´s also a possibility of both sizes being the same. So there's not really a landscape or portrait in this case.