In a self-important, gossipy town like Washington, D.C., where almost no conversation is safe from leaks, there is one bastion of secrecy on Capitol Hill.

The Chowder and Marching Club is kept so far under wraps that members are reluctant to even acknowledge its existence. “I don’t even know what you are talking about,” said U.S. Rep. Ileana Ros-Lehtinen of Florida.

When...