Why is java.exe not visible from cygwin, but is visible from command-prompt?

In x64 editions of Windows 8/7/Vista, contents of C:\Windows\System32 folder depend on the bitness of the application .

For 64-bit applications (such as Windows Command Prompt, cmd.exe) there's no filesystem redirection:
C:\Windows\System32 (in software) -> C:\Windows\System32 (on disk).
So, C:\Windows\System32 folder contains 64-bit DLL and EXE files, from the point-of-view of 64-bit application.

For 32-bit applications (such as bash.exe from Cygwin) which runs via WoW64 subsystem on Windows x64, there are two redirections in the filesystem:
C:\Windows\System32 (in software) -> C:\Windows\SysWOW64 (on disk).
C:\Windows\Sysnative (in software) -> C:\Windows\System32 (on disk).
So, C:\Windows\System32 folder contains 32-bit DLL and EXE files, from the point-of-view of 32-bit application.

If you want to access 64-bit applications and DLL files (such as java*.exe) from 32-bit applications (such as Cygwin's bash.exe), you need to use C:\Windows\Sysnative path, or /cygdrive/c/Windows/Sysnative in Cygwin notation.

(this is an updated and extended answer)