Change entered values to positive or negative values

Actually, it probably has to be

-1 * [My Data entry field]

But A formula may not contain a reference to itself directly or indirectly through another formula?

Are you trying to change the sign on a data entry field or a formula field.  if its a data entry field then you will nee to create a new field

Its a data entry field. I've asked why cant the user just enter the information correctly, but they want to try and ensure data integrity. :) 

It is a confusing request. I'm sure that I could find a solution, but the idea of flipping the sign on whatever the user enters seems so foreign to me that I hesitate to suggest a solution.

Are you saying that if the user enters a positive number you want that data entry to be changed to negative and vice versa?

I would have to understand the use case for this need.

Yes. That is exactly what they are asking for. I know its confusing and strange, which is prolly why I haven't been able to figure out a solution. 

I'm not going to contribute to that madness.  :) 

Imho it is too dangerous to change the user's actual input.  I suggest a formula field to be used which calculates the opposed signed result of what the user entered, and show that on reports and in view mode.

I say dangerous in the sense of never really being sure what the user entered and if an automated process flipped the sign.  Then what happens if the user edits the record, do we flip the sign again?

Agreed. What about just being able to keep only positive numbers. If a user enters a negative number it is automatically "flipped" to positive. If they enter a positive it stays positive?

If you make a field with a formula of ABS([my data entry field]). Call it [Positive Field],

and you put that field on the form (for now, we can hide it later) then you should be able to have a form rule that says

When the record is saved
 and [my date entry field] is not equal to the value in the field [positive field]

Change the value of [my data entry] field to the value in the field [Positive value]