Windows – ‘impdp’ is not recognized as an internal or external command, operable program or batch file

I can see impdp.exe in E:\Oracle\product\11.2.0\dbhome_1\bin, and that folder is in the PATH. However when I open a CMD window and try to run impdp, I get this error:

'impdp' is not recognized as an internal or external command, operable
program or batch file

I can verify the PATH is correct, because I can run the imp command from the same folder. Using the full path name to impdp does not help. I can run impdp by double clicking it in Windows Explorer, but that does not let me specify any command line arguments I need.

I've checked both imp and impdp with Dependency Walker, and they are both missing the same demand loaded DLLs. This is a Windows Server 2003 64-bit machine, if that makes any difference. I get the same results in both 32-bit and 64-bit CMD windows.