We analyze the physics of self-bound droplets in a doubly dipolar Bose-Einstein condensate (DDBEC) composed by particles with both electric and magnetic dipole moments. Using the particularly relevant case of dysprosium, we show that the anisotropy of the doubly-dipolar interaction potential is highly versatile and nontrivial, depending critically on the relative orientation and strength between the two dipole moments. This opens novel possibilities for exploring intriguing quantum many-body physics. Interestingly, by varying the angle between the two dipoles we find a dimensional crossover from quasi one-dimensional to quasi two-dimensional self-bound droplets. This opens a so far unique scenario in condensate physics, in which a dimensional crossover is solely driven by interactions in the absence of any confinement.Ultracold atoms are in the forefront for probing quantum many-body physics [1][2][3] due to the ability to manipulate them using external fields and engineering both short [4] and long-range interatomic potentials [5,6]. Among them, dipolar gases [7-9] gained a lot of attention since the first realization of the dipolar Bose-Einstein condensation (DBEC) of chromium atoms [10,11]. The quest for systems with higher dipole moments paved the way for polar molecules [12], ultra cold Rydberg atoms [13], erbium [14] and dysprosium (Dy) [15] BECs. For the first two cases, it is the electric dipole moment whereas in atomic condensates it is magnetic in nature. Due to the potential applications in quantum simulation [16], computing [17], tests of fundamental symmetries [18] and tuning of collisions and chemical reactions [19], a growing interest is focusing on particles possessing both electric and magnetic dipole moments [20][21][22][23][24]. Although these studies have been mostly restricted to paramagnetic polar molecules [25][26][27][28], a recent proposal has discussed the feasibility of realizing an electric, in addition to the magnetic, dipole moment in Dy atoms [20]. The latter is particularly important step due to the experimental feasibilities in atom-based setups, and opens intriguing questions about the effect of doubly dipole-dipole interactions (DDDIs) in quantum gases.Due to its anisotropic and long-range nature, dipole-dipole interaction (DDI) results in novel phenomena in DBECs [7][8][9], including anisotropic superfluidity [29][30][31], roton-like excitations [32,33], quantum droplets [34-37] and transient supersolid states [38][39][40]. Quantum droplets have been the focus of large attention, since they result from the stabilization due to quantum fluctuations of an otherwise collapsing DBEC.In this letter, we discuss the novel physics of condensates of particles with both magnetic and electric dipole moments. Focusing on the particularly relevant case of Dy atoms, we show that the combined DDDI exhibits a non-trivial anisotropy, which may be readily controlled by means of the relative orientation and strength of both dipole moments. This nontrivial anisotropy results in an intriguing physi...