\[P(x,y), \: Q(x,y)\]
defined on a region \[R\]
with boundary \[C\]
;\[\oint_C P \; dx + Q \: dy = int \int_R (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) \: dx \: dy\]
Interchanging
\[P,Q\]
and \[x,y\]
respectively gives\[\oint_C Q \; dy + P \: dx = int \int_R (\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x} ) \: dx \: dy\]
The Left hand side is the same as before, but the right hand side is reversed in sign, which amounts to reversing the orientation of the curve.
With
\[\mathbf{a} = P \mathbf{i} + Q \mathbf{j}\]
and \[\mathbf{r}= x \mathbf{i} +y \mathbf{j} \rightarrow d \mathbf{r}= dx \mathbf{i} +dy \mathbf{j}\]
\[\oint_C \mathbf{r} \cdot d \mathbf{r} = - \oint_{-C} \mathbf{r} \cdot d \mathbf{r}\]