Context. Studying outflows is important, as they may significantly contribute to angular momentum removal from a star-disc system and thus affect disc evolution and planet formation. Aims. To investigate the different outflow components, including the collimated jet, wide-angled molecular outflow, and outflow cavity, of the Class I HH 46/47 outflow system, we focused on their kinematics. Methods. We present near-infrared (NIR) K-band integral field observations of the blueshifted HH 46/47 outflow base obtained using VLT/SINFONI with an angular resolution of 0.″81. Our analysis focuses on [Fe II], H₂ 1‑0 S(1), and Br-γ emission. We employed a wavelength recalibration technique based on OH telluric lines in order to probe the kinematics of the wide-angled flow with an accuracy of ∼1 km s^‑1–3 km s^‑1. Results. We confirmed a velocity gradient of ∼ 10 km s^‑1 transverse to the outflow direction in the wide-angled H₂ outflow cavity. We find that the H₂ cavity peaks at radial velocities of ∼‑15 km s^‑1 to ‑30 km s^‑1, and that the atomic jet peaks at v_rad ∼ ‑210 km s^‑1. The outflow exhibits a layered structure: The high-velocity [Fe II] and Br-γ jet is surrounded by a wide-angled H₂ outflow cavity that is in turn nested within the continuum emission and CO molecular outflow. The continuum emission and H₂ outflow cavity are asymmetric with respect to the jet axis. Conclusions. We propose that the origin of the asymmetries and the velocity gradient detected in the wide-angled H₂ cavity is due to a wide-angled outflow or successive jet bowshocks expanding into an inhomogeneous ambient medium or the presence of a secondary outflow. We eliminated outflow rotation as an exclusive origin of this velocity gradient due to large specific angular momenta values, J(r) ≈ 3000‑4000 km s^‑1 au, calculated from 1″ to 2″ along the outflow and the fact that the sense of apparent rotation we detected is opposite to that of the CO envelope emission. The observations reveal the complexities inherent in outflow systems and the risk of attributing transverse velocity gradients solely to rotation. ⋆ Based on observations collected with the VLT operated by the European Southern Observatory under ESO program ID 0102.C-0615(A).