Volborthite Cu3V2O7(OH)2·2H2O is a copper mineral that materializes a two-dimensional quantum magnet comprising a kagome net of spin-1/2 Cu2+ ions. We prepared single crystals of volborthite using hydrothermal conditions and investigated their crystal structures and magnetic properties. Unusual orbital "switching" and "flipping" transitions were observed: in the former type of transition (switching), the Cu 3d orbital occupied by an unpaired electron changes between the d(3z2-r2) and d(x2-y2) types, and in the latter type of transition (flipping), the d(x2-y2)-type orbitals change their directions. Their origin is ascribed to variations in the orientation of water molecules in the gap between the kagome layers and the accompanying changes of hydrogen bonding. These orbital transitions dramatically modify the magnetic interactions between Cu2+ spins, from the anisotropic kagome type to the formation of spin trimers over the kagome net. The effective spin 1/2 generated on the trimers exhibits a frustrated magnetism, resulting in a rich phase diagram in the magnetic fields. Volborthite is a unique compound showing an exceptional interplay between the orbital and spin degrees of freedom.