In this article, a wavelet entropy, which behaves as a shifted version of the standard wavelet $(q,q^{\prime })$-entropy of fractal signals, is presented. The shifted wavelet $(q,q^{\prime })$-entropy is obtained by computing the standard $(q,q^{\prime })$-entropy functional on a weighted relative-wavelet-energy (RWE) representation of fractal signals; it is shown that the weight within the RWE plays the role of a shifting factor in the characteristics of the standard wavelet $(q,q^{\prime })$-entropy. Therefore the shifted wavelet $(q,q^{\prime })$-entropy relocates the wavelet entropy values to any point of the fractality index range, which allows us to analyze a wide variety of fractal signal families thus improving on previously proposed entropies in the literature. Information planes for these entropies are obtained using different shifts and values of parameters $q$ and $q^{\prime }$, which allow us to highlight the potential applications for a fractal signal analysis. Moreover, an experimental study using synthesized exact fractal signals shows that the shifted wavelet entropy can classify stationary long-memory signals from short-memory ones and can also be used to differentiate other fractal signal families.