The landscape of ultrafast structured light pulses has significantly advanced thanks to the ability of high-order harmonic generation (HHG) to translate the spatial properties of infrared laser beams to the extreme-ultraviolet (EUV) spectral range. In particular, the up-conversion of orbital angular momentum (OAM) has enabled the generation of high-order harmonics whose OAM scales linearly with the harmonic order and the topological charge of the driving field. Having a well-defined OAM, each harmonic is emitted as an EUV femtosecond vortex pulse. However, the order-dependent OAM across the harmonic comb precludes the synthesis of attosecond vortex pulses. Here we demonstrate a method for generating attosecond vortex pulse trains, i.e., a succession of attosecond pulses with a helical wavefront, resulting from the coherent superposition of a comb of EUV high-order harmonics with the same OAM. By driving HHG with a polarization tilt-angle fork grating, two spatially separated circularly polarized high-order harmonic beams with order-independent OAM are created. Our work opens the route towards attosecond-resolved light-matter interactions with two extra degrees of freedom, spin and OAM, which are particularly interesting for probing chiral systems and magnetic materials.