Davidson wrote on Jan 22^nd, 2008 at 3:39pm:

I am in a similar position - made it to Phase II and was told that I failed one section of the polygraph. I sent a letter in requesting a second polygraph, and was told I "did not meet the criteria required and will not be afforded further consideration..." Is there an appeals process? Can you offer some suggestions as to the content of the appropriate letter? Thanks.

I was in a similar position as well, and while Executive Order 12968 (quoted later) does "permit" for a further appeal, it may not be "worth it" to continue.  I decided to give up at the denial of a retest.

"(6) provided an opportunity to appeal in writing to a high level panel, appointed by the agency head, which shall be comprised of at least three members, two of whom shall be selected from outside the security field. Decisions of the panel shall be in writing, and final except as provided in subsection (b) of this section; and

(7) provided an opportunity to appear personally and to present relevant documents, materials, and information at some point in the process before an adjudicative or other authority, other than the investigating entity, as determined by the agency head. A written summary or recording of such appearance shall be made part of the applicant's or employee's security record, unless such appearance occurs in the presence of the appeals panel described in subsection (a)(6) of this section."