Performing reverse clip in ArcMap?

Use the Erase (Analysis) Tool:

Another option here would be to perform a union and then delete the inner feature. After the union, select features from the initial layer and use the delete feature tool to remove the selected polygons.

Since Erase (as @Jens linked) only is available with an Advanced license, you can download ET Geowizards. It can be installed as an Arcmap toolbox.

Although you have to pay for the full suite, there's a free part of the program and the Erase function is included there (Overlay group).