Tom and his wife are introduced as thoroughly unlikable people, and their marriage seems to bear more resemblance to the relationship one would expect to see between sworn enemies. As a fulfillment of their description as misers, Tom and his wife are said to cheat each other at any opportunity.
The term "cheat", as used in the story, probably has no connection to the modern slang that refers to infidelity. Rather, Tom and his wife seem to treat their marriage almost like an antagonistic game, and cheating in this context refers to the ways in which they defy the standard behaviors and expectations of a marriage, consumed as they are with greed. It is mentioned that they can barely resist fighting over things which, under any other circumstances, would be considered common property. Thus, they steal physical property from their common ownership, and also metaphorically steal any enjoyment or comfort that the other might reasonably find in the marriage.
This helps to explain several subsequent elements of the story. For example, it makes sense of the harsh manner in which Tom's wife greets the news of his meeting with Old Scratch, and her intention of taking the treasure for herself. It also clarifies Tom's motivation in spiting her through his sudden reversal and apparent lack of interest in going through with the bargain. Finally, it explains why Tom hardly mourns his wife's disappearance.