I use the free program Image Magick for command-line processing of images in Cygwin on Windows. The used version is 7.0.10. For convenience I define the shell variable in Cygwin:

export IM="/cygdrive/c/Programs/ImageMagick-7.0.10-Q16-HDRI/magick.exe"

To get the basic information (e.g. size) about an image type in the Cygwin shell:

$IM identify image.jpg

To get the complete information including EXIF metadata use:

$IM identify -verbose image.jpg

Let us consider the image "Bridge":

$IM identify bridge.jpg
bridge.jpg JPEG 684x456 684x456+0+0 8-bit sRGB 216961B 0.016u 0:00.006

The size of the image "Bridge" is 684x456 and we interpret it as "horizontal_width x vertical_height".

Let us consider the image "Tower":

$IM identify tower.jpg
tower.jpg JPEG 684x456 684x456+0+0 8-bit sRGB 216961B 0.016u 0:00.006

The size of the image "Tower" is 684x456 but is this "width x height" ? It is in fact width x height of the stored image. This image was taken with the camera in the "vertical" position so the "width" is the vertical dimension of the dispalyed image. The information about the orientation of the camera is fortunately recorded in the "EXIF" metadata tag "Orientation" of the photo. Most applications and browsers display the image rotated according to "Orientation". When the application GIMP opens an image in a rotated orientation it offers the user the option to keep or to adjust the orientation. If we choose "Keep Orientation" GIMP displays: We obtain the "Orientation" and "Size" by issuing:

$IM identify -verbose tower.jpg | grep "Orient\|Geom" 
Geometry: 684x456+0+0
Orientation: RightTop

Compare this with:

$IM identify -verbose bridge.jpg | grep "Orient\|Geom" 
Geometry: 684x456+0+0
Orientation: TopLeft

The Exif tag "Orientation" has one of 8 values that are explained below.

$IM identify -format '%w %h %[orientation]' tower.jpg
684 456 RightTop