If I have some two form $\omega = f(x,y) dx \wedge dy$ and I want to integrate it over some rectangular region $R$ then I have the following: $$ \int_R \omega = \int\int f(x,y) dx dy$$ but $\omega$ is also equal to $-f(x,y) dy\wedge dx$ so we also have $$\int_R \omega = \int\int -f(x,y) dy dx$$ but by Fubini's theorem this second integral on the right is the same as $$-\int\int f(x,y) dx dy$$
How does this make sense? The integral of $\omega$ is plus or minus the integral on the right?
The formula that you're using, $$ \int_R f(x^1,\dots,x^n)\, dx^1\wedge\dots\wedge dx^n = \int\cdots\int f(x^1,\dots,x^n)\, dx^1\dotsm dx^n, $$ only holds when the coordinates $(x^1,\dots,x^n)$ in that order are positively oriented for the manifold in question. On $\mathbb R^2$ with its standard orientation, $(x,y)$ is a positively oriented coordinate chart, but $(y,x)$ is not.
The apparent paradox that you noticed is exactly the reason why differential forms can only be integrated over an oriented manifold, and the integral has to be calculated in oriented coordinate charts.