The local stellar halo of the Milky Way contains the debris from several past accretion events. Here we study in detail the structure and properties of nearby debris associated with the Helmi streams, originally identified as an overdensity in integrals of motion space. We use 6D phase-space information from Gaia EDR3 combined with spectroscopic surveys, and we analyse the orbits and frequencies of the stars in the streams using various Galactic potentials. We find that the streams are split into substructures in integrals of motion space, most notably into two clumps in angular momentum space. The clumps have consistent metallicity distributions and stellar populations, supporting a common progeny. In all the realistic Galactic potentials explored, the Helmi streams stars are on different orbital families and spread across multiple resonances. For example, ~40% of the streams stars populate the Omega_z : Omega_R ~ 1:2 resonance tightly, while the remainder depict a more diffuse distribution close to Omega_z /Omega_R ~0.7. At the same time, the reason for the substructure in angular momentum space appears to be due to a Omega_z : Omega_phi resonance close to the 1:1. There is also substructure in velocity space which can be related to the presence of these multiple resonances. Our findings suggest that the structure of the Galactic potential leaves a clear imprint on the properties of phase-mixed debris streams, and as a consequence, that these are more complex than considered thus far. It is however not clear why the particular resonances identified are populated as observed.