My best advice is to look elsewhere for a job.  I know it sucks to have to do that, but unfortunately they have already branded you and made their own assumptions about your character.  Fight the failure if possible, because if you apply anywhere else it may come back to bite you in the ass.  I don't know why, but other departments (especially the ones that don't polygraph) seem to automatically see a polygraph failure or inconclusive outcome as a problem and usually will not hire that applicant.  Even when your background investigation provides nothing to disqualify you they will use the polygraph from the other department to disqualify you under their "moral character" category and claim that you are hiding something.  Are you even sure you really want to be a cop?  With all the lawsuits and video camera carrying civilians even a perfect arrest can come across as you using "excessive force"  and to avoid any community backlash your department may be likely to fire you.  I will be honest with you, I love working in law enforcement and I couldn't really imagine doing anything else, but the bureaucracy, politics, and never ending fear of losing your job or being sued for doing your job really sucks.
Either way good luck and I do feel your pain in the polygraph experience.

I suggest keeping the tone of your "scorching" letter firm but civil. Explain that you did not use countermeasures, and request a third polygraph examination. At present, there is no other appeal process. The chief of the L.A. Public Safety Employment Division is Phyllis M. Lynes. Her address is:

PHYLLIS M. LYNES
CHIEF, PUBLIC SAFETY EMPLOYMENT DIVISION
700 EAST TEMPLE STREET
LOS ANGELES CA 90012

You can also e-mail her at plynes@per.lacity.org or call her at (213) 847-9717.

I just received the results of my 2nd poly. I Failed. I don't get it. I was asked all the standard questions and then again. I was told my breathing was atrosious, I was also accused of using countermeasures( I had no clue to what this meant). This examiner even offered me a third opportunity if I would cop out to the accusations. He suggested that I was squeezing my anal muscles or had been given bad advise by my police officer friends who are on the job now. The only answer they can give me is that my application information and results of my poly do not match.   I am preparing my scorching letter now to the new Chief and Personnel Dept. I scored a 91 on my oral, No prior criminal history or Police Contact / FI, Perfect Credit, Stable work history. WHAT DOES IT TAKE? IS THERE A LIGHT?