Calling async methods from non-async code

I’m in the process of updating a library that has an API surface that was built in .NET 3.5. As a result, all methods are synchronous. I can’t change the API (i.e., convert return values to Task) because that would require that all callers change. So I’m left with how to best call async methods in a synchronous way. This is in the context of ASP.NET 4, ASP.NET Core, and .NET/.NET Core console applications.

AspNetSynchronizationContext

Trying to use new C# 5 async model it was surprising to me AspNetSynchronizationContext is an internal class (as well as AspNetSynchronizationContextBase base). Thus undocumented. But it’s essential to know what it does when utilizing async/await feature within your ASP.NET code. Am I correct that
It does guarantee your continuations will get the same HttpContext.Current as original callers?
It does not guarantee the continuations will execute on the same thread as the callers?