Ticket Iteninery was correct but information that was sent to airlines who printed ticket incorrect.

I purchased 2 tickets from Maui to Bangkok from Cheapoair over the phone for $2,370.00 USD, I went over the spelling of our names 3 times and the agent on the line read it back to me correctly.  I went directly to my computer when I got home and printed out the itinerary and double checked the spelling on both names along with dates and times of travel.  All was correct.  On the date of travel, we went to the airport, had the Airline Representitive print our ticket to Japan (that was where we had a layover) and boarded our plane thinking everything was fine.  Once in Japan we went to the counter and got our next ticket and showed our passport.  The Airline Representative showed us that both of our names were spelled incorrectly.  I showed her our itinerary with the correct spelling and she explained that the information that her airlines had received from Cheapoair was spelled incorrectly even though our itinerary was spelled correctly. Thank goodness she let us board the plane and we made it to Bangkok.  Once in Bangkok I called Cheapoair and explained this all to them.  They told me to fax them a letter explaining with my passports and itinerary with the correct spelling.  I did this and the next day got a response from Cheapoair.  Their response said I should never have flown with misspelled tickets and that they would not correct their mistake nor refund my money. (even though I showed them my correct itinerary)  I am very dissatisfied with this company and will never use them again.  This was 100% their fault and they took no responsibility.  We were now stuck in Bangkok and had to buy two tickets to get home at the cost of $2,200.00 USD.