As you know, C and C++ are sister languages that have a lot in common, but that drifted quite apart over the years. In general, neither code of one language can be compiled as the other language, there are too many major and minor twists that make this impossible. Not only are there syntax differences between the two, some common syntax can actually have diverging semantics. So generally, it makes no sense to compile C code with a C++ compiler, and you should look with suspicion at any code or programmer that claims to do so.
Where C and C++ usually agree, though, is on the ABI, the application binary interface, so data structures and functions of one language can be used by the other to some extent. C and C++ also kept a sufficiently wide intersection in there respective specification of interfaces, such that one header file can be used from both.
In this post I try collect those parts that are in that intersection, and I propose some coding style that should accommodate both worlds suitably well. But as my personal history goes, this will merely be a POV of a C programmer that wants to provide interfaces for C++.