Any object or function that resides in 3-dimensional space can be rotated. Either actively by moving the parts a few degrees at a time about a given axis, or passively by rotating the coordinate system about a given axis. After 360 degrees the object and coordinate system will necessarily be in an identical position. A 360 degree rotation is a symmetry operation - it must be. Of course if the object has additional symmetry then a rotation by a smaller angle could also be a symmetry operation.
But for fermions a 360 degree rotation is not necessarily a symmetry operation, but a 720 degree rotation is. This extraordinary result is both an experimental fact and is correctly shown by the maths of quantum theory. Fermions are described as spin-half particles and quantum theory needs to represent them by a spinors (rather than vectors or scalars). The maths works out because wavefunctions are not real objects (they are a sort of probability function) they are only defined up to a global phase factor so that Psi and -Psi both describe the same state. This ambiguity in the phase permits unusual behaviour under rotations. There are deep mathematical reasons why a 720 degree rotation must still be a symmetry operation.
Although the maths works out the correct results, it fails to explain why an electron (for example) needs to be described by a spinor. An electron is real, if it really exists in 3D space then why is a 360 degree rotation not a symmetry operation?
There are many popular descriptions of spin-half like properties. They all have a common feature, part of a flexible object is rotated while another part is held fixed. As you travel from the fixed point to the rotated end, parts of the object undergo a continuous range of rotations. In 3D a 360 degree rotation performed this way, twists the object (as it would in 2D). A 720 degree rotation appears to twist the object even more: in 2D it is indeed more twisted. But in 3D it is possible to straighten out a flexible object when one end is rotated 720 degrees and the other is fixed. In all popular examples a 360 degree rotation of the object is still a symmetry operation if applied to the whole object.
What prevents a 360 degree rotation of an electron from being a symmetry operation. How could one part be held fixed? My paper Spin half from classical general relativity explains how.