The couple, a man and a woman, took several bags of dirty washing for a service wash at Leeds Road Laundry at around 10am on Monday 17 June 2013.
After leaving their laundry, which included mats from a caravan, they returned to collect it at around 2pm that afternoon.
While the laundrette staff were working in the back of the shop, the couple took their clean and dry washing without making any attempt to pay.
Officers are appealing to anyone who can help to identify the suspects to come forward with information.
The man, who gave his name as Brucie, is described as white, aged in his 30s, around 5ft 9 to ft 11in tall, with a stocky build and short hair. He was wearing pale blue polo shirt with a dark collar and Adidas tracksuit bottoms.
The woman was white, with dyed blonde hair. She was wearing white shorts and a yellow top.
It is believed that the couple left the scene in a black Ford S-Max.
Anyone who can help officers with their enquiries is urged to contact North Yorkshire Police on 101 – select option 2 – and ask for PC Andy Hanson or Harrogate police.
Alternatively, information can be emailed to firstname.lastname@example.org
If you wish to remain anonymous, Crimestoppers can be contacted on 0800 555 111.
Please quote reference number 12130098776 when passing information about this incident.