Aims: In this paper we analyze the spectra of a sample of asteroids in cometary orbits (ACOs) in order to understand the relationship between them, the Jupiter family comets (JFCs), and the outer main belt populations, such as Hilda, Trojan and Cybele asteroids. Methods: We obtained visible (0.55-0.90 μm) and/or near-infrared (0.8-2.3 μm) spectra of 24 ACOs using 3 telescopes at the “Roque de los Muchachos” Observatory (La Palma, Spain). Using this data, we derived the taxonomic classification of the asteroids. As most ACOs present featureless spectra (B-, C-, P-, D-type) we also derived their spectral gradient (S'). Considering also published spectra of ACOs we correlated S' with orbital and dynamical parameters and obtained the cumulative distribution of S', and compare it with that of other related populations. Results: We present visible and/or near infrared spectra of 24 ACOs, most of them (21) presenting featureless spectra. After including the spectra of other ACOs already published, we analyzed a total of 41 objects, 34 of them having featureless spectra like the spectra of comet nuclei and outer main-belt asteroids. We also noticed a significant difference in the taxonomic distribution of the ACOs in Near-Earth orbits (q<1.3 AU) and those that are not (q>1.3 AU), indicative of different source/transport mechanisms. About 35% of the ACOs in the NEO population analyzed have spectra that present the typical silicate absorption bands at 1 and 2 μm (S- and V-type), while only 1 of the 24 ACOs in the non-NEO population (about 4%) is S-type and the other 23 have a featureless spectrum. Thus the NEO sub-population of ACOs is composed of a significant fraction of asteroids scattered from the inner main-belt. We didn't find any subtle features in the 0.5-2.0 μm spectral region of featureless ACOs that can be used to discriminate wheather an ACO comes from a cometary or an asteroidal origin. The analysis of the spectral gradient shows that ACOs present an interesting, and significant, anti-correlation between the Tisserand parameter and the “spectral gradient”, meaning that the reddest objects have the lower Tisserand parameter (i.e., higher chance of a cometary origin). Finally, we obtained the S' cumulative distribution for ACOs. This distribution is “bluer” than that of comet nuclei, Damocloids and outer main-belt population of asteroids, indicative of a significant “contamination” of asteroids scattered from the inner main-belt.