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Let $X$ be an interpolation space between $L^1(0,1)$ and $L^\infty(0,1)$. We consider the Banach algebra $W^{1,X}$ of all absolutely continuous functions on [0, 1] with derivatives in $X$. Let $T$ be a bounded weighted composition operator on $W^{1,X}$. The study of essential spectra of $T$ can be reduced to that of some weighted composition on $X$. That allows us to give a simple condition for essential spectra of $T$ to be rotation invariant. We obtain a complete description of the spectrum and essential spectra of $T$ in the case when the Boyd indices of $X$ are equal and strictly between 0 and $\infty$, and $T$ is an invertible operator.