By default Visual Studio organizes my project like this:
It turns out the issue was that the "Show All Files" button does not appear if you are selecting the Solution itself. If you select an underlying Project, or any of the folders/files in the project, the button will display:
And here is the Solution Explorer view after clicking the button:
Note: see also this post for a solution for Visual Studio 2012, which didn't solve my problem but is closely related.
It didn't work for me, but if it's all about including an existing file to the project, you can copy the file or folder from the Explorer and right click the parent folder (the same folder it exists in) in Visual Studio Solution Explorer and click paste.