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Are there any general conditions to use to find a vector field f(x,y) that is a gradient field and f(-y,x) is also a gradient field. It seems to me like if their second partial derivatives are zero then this is true or at least I haven't find an exception to that yet.

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    The normal of a vector field $(f(x,y), g(x,y))$ is $(-g(x,y), f(x,y))$, not $(f(-y,x), g(-y,x))$ as you have written.2012-01-06
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    Indeed, your question and its title ask different questions!2012-01-06

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