How can I reflect a transparent background? I'd like to make a glossy surface which reflects objects above it, but that also reflects a transparent background in the final render.

So in effect, the final render would contain only the reflections, not the surface itself.

Is this possible please?
You need to use a Transparent shader, and a Glossy BSDF shader, combined with an Add Shader, not a mix shader. 

Using the Add shader gives a perfect result:

Nodes:



Result:



Although you should normally use a Mix shader (greater realism) this is a perfect use case. 

Here is a comparison of the two methods: