When you are in an orthogonal view along one of the axes, the reflection image normally displayed behind the Material View is suppressed or averaged or something. As soon as you get off-axis it is back.
Is there a way to get a better representation of the reflection image while in orthogonal views?
Not sure what you meant: Do you know: Prefences → Lights → Studio Lights ?
Or did you tried Matcaps … 
okay in ortho ther is lesss shading… but ther relection is there.
This is in SHADING MODE:
AXIS VIEW:
OFF AXIS:
I think it’s just a highly averaged version of the reflection images, but it seems like it should be a bit better.
The real issue seems to be that if you use NUMPAD 1 it automatically puts you in orthogonal mode, because NUMPAD 5 will get you back to Perspective and the bkgnd image. So, defeating this automatic switch (grrrrr) would be the answer.
Hmmm there seem to be not much to reflect here… anyway NUM 1 is ORTHO FRONT , NUM 3 orto right …
Try Num 0 Camer (persp or ortho like camera setting ) and NUM . User Persptive
