Cube is standing on mirror plane and Lamp without diffuse lighting produces strange specularity in mirror's reflection. It is strange, because actual object doesn't have such specularity. Is it normal?
The reason you can't see any specular reflections on the cube is because they are just that, reflections. Since the cube's faces are flat planes, you will only see specular reflections if you are at the opposite angle of your light source.
The reason they show up in the reflection is because the light is bouncing off of the face of the cube, (just away from the view/camera which is why you can't see it) and then reflecting off of the mirror.
I tested this on some other objects as well, all using the same materials. The mirror reflects the specular light based on where it reflects off of the object and then off of the mirror.
What it comes down to (as far as I understand) is that we expect the light to behave like diffuse light (the mirror reflects what we see exactly) but instead the light bounces off of the object, then off of the mirror and into the camera, we just can't see when it bounces off of the object since we are at the wrong angle.
You can test this in the real world as well, by using two perpendicular mirrors and a light source. When set up like you have it, you will only see the light source in the horizontal mirror.