Why is there a clear mirror image off a rough surface at shallow angles? The mirrors we use everyday have a highly reflective smooth surface, so that we can see a clear image.  But I found the same happens on a rough (not so smooth) surface, though only at shallow angles.  For example

this is the logo of physics stackexchange, reflected from the back of an ipad.
Considering that aluminum may contributed to reflection, I tried a different material:

this time a piece of printer paper.  Though not as clear, there is still some visible pattern resembling the logo.
The same effect can be observed on highways, where distant vehicles have a mirror image off the road; I do not have a picture at hand however.
So, why is this happening, and why only on shallow angles?
 A: The reason for this is 2-fold:
 1. Surfaces reflect better at almost parallel incidence
Let's focus on dielectric materials. Then this statement is a simple consequence of Fresnel's laws of reflection. Here are two reflection curves plotted against incidence angle from the wikipedia article (angle is measured from the normal to the reflection surface, so $90°$ means almost parallel to the surface):


As you can see in both cases (glass to air and air to glass) the reflection goes to one when the angle of incidence goes to $90°$.
This is fact is indeed used e.g. for X-rays, which essentially pass through everything at normal incidence, but are reflected well at "grazing incidence".
Here is an intuitive and not very precise reason for why this happens: when you shoot at something at almost parallel to the surface, then the component of the wave vector perpendicular to the surface becomes very small. So it has less energy perpendicular to the surface to punch through.
Just so that people don't kill me for the last sentence: this is sort of how Fresnel's equations are derived. The parallel wave vector is conserved and the perpendicular one transforms accordingly to match the frequency (energy) inside and outside the material. So that intuitive reason actually makes sense even though it sounds stupid.
 2. Small incidence angles smooth over roughness
Imagine you look at the ocean. When you look from above, you will be able to see foamy wave tops all over the place. But when you look parallel to the surface, you will only be able to see one wave top. The other ones are all behind it, you might be able to see a few more a long way away, just peaking over the first crest.
That is intuitively why low angles smooth over rough surfaces. In a more rigorous manner we can state it in terms of the perpendicular component of the wave vector gain. Since it is smaller, the corresponding wavelength $\lambda = \frac{2\pi}{k}$ is much larger. The reflected beam can therefore only resolve much larger distance scales and roughness on small scales is smoothed over. I'd love to have a picture for this, but unfortunately I can not find a good one right now.
A: I'm wondering whether it's something like this… On a small scale the rough surface consists of hills and valleys. But the hilltops and valley bottoms are all (at least over a small area) parallel to the general plane of the surface, so specular reflection (angle of incidence = angle of reflection) from those bits of surface will re-enforce. Now, according to Fresnel's laws of reflection, the reflected intensity from the surface of a dielectric medium rises to a maximum at grazing incidence – the 'shallow angle' you mention.  
A: The rough surface (matte surface) is like lots of tiny reflective surfaces laid out in peaks and valleys but at random angles.
Imagine this
If a small column of light with a cross section area (one pixel from an image) strikes a matte surface at 45' and you observe this at a reflective angle.
The light will reflect off at random angles and only a small percentage (from that pixel)from the small area where the column struck the surface will make it to your eyes.
You will perceive this as a rough surface with no definitive reflective image.
Now that same column of light (from the same pixel) strikes the surface at say 5' from the surface (shallow angle) and you observe it at a reflective angle, where that same column of light strikes the surface the area is larger lets say a strip from one side towards the other.
You will observe that strip as being short due to your shallow viewing angle.
At every point along that strip the light will be reflected off at a random angle and only a small percentage will make it to your eyes.
But there is a lot more area and at each point along the strip another small percentage of light will make it to your eyes from the original pixel.
These will all add up for you to recognize a bright pixel.
A: Good reflectors are usually materials that are conductors.
If you see an image only at particular angles then either the reflectivity is low and the direct light brightness mask the image or somehow predominantly this is a preffered direction (in random pattern this is unlikely).
The images of vehicles that you are describing are likely mirages where the refractive index change in height bends the light rays and creates an off-setted image.
A: The roughness of a surface degrades the quality of the reflection as the wavefront suffers a position dependent phase delay which is comparable to the wavelength of the light. The phase delay is simply caused by different path lengths from rays reflecting from different points with different heights. 
The main point though is that this path length difference is proportional not only to the roughness (height of the imperfections) but also to the sin of the angle of incidence.
So as you approach grazing incidence the path difference decreases and you get a better reflection.
I think the following paper explains this well:


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*Fakhruddin, H. Specular Reflection from a Rough Surface. Phys. Teach. 41, 206 (2003).
