Context. The study of protoplanetary disks is fundamental to understand their evolution and interaction with the surrounding environment, and to constrain planet formation mechanisms. Aims. We aim to characterise the young binary system HD 34700 A, which shows a wealth of structures. Methods. Taking advantage of the high-contrast imaging instruments SPHERE at the VLT, LMIRCam at the LBT, and of ALMA observations, we analyse this system at multiple wavelengths. We study the morphology of the rings and spiral arms and the scattering properties of the dust. We discuss the possible causes of all the observed features. Results. We detect for the first time, in the Hα band, a ring extending from ~65 au to ~120 au, inside the ring which is already known from recent studies. These two have different physical and geometrical properties. Based on the scattering properties, the outer ring may consist of grains with a typical size of a_out ≥ 4 µm, while the inner ring has a smaller typical size of a_in ≤ 0.4 µm. Two extended logarithmic spiral arms stem from opposite sides of the disk. The outer ring appears as a spiral arm itself, with a variable radial distance from the centre and extended substructures. ALMA data confirm the presence of a millimetric dust substructure centred just outside the outer ring, and detect misaligned gas rotation patterns for HD 34700 A and B. Conclusions. The complexity of HD 34700 A, revealed by the variety of observed features, suggests the existence of one or more disk-shaping physical mechanisms. Our findings are compatible with the presence inside the disk of an as of yet undetected planet of several Jupiter masses and the system interaction with the surroundings, by means of gas cloudlet capture or flybys. Further observations with JWST/MIRI or ALMA (gas kinematics) could shed more light on them.