a polar crystal class
A polar crystal class is one whose symmetry leaves a unique direction singled out, a 'special axis' that points one way and is not equivalent to pointing the opposite way. Along that axis the crystal has a built-in electric lopsidedness, a permanent separation of positive and negative charge, called a spontaneous polarisation. Picture an arrow frozen into the crystal that no symmetry operation can flip end-for-end.
The test is precise: a class is polar if it has no symmetry operation that reverses the special direction, no inversion centre, no mirror perpendicular to it, no perpendicular 2-fold axis. Exactly 10 of the 32 crystal classes qualify (1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6mm). These 10 are the pyroelectric classes: heat such a crystal and the polarisation changes, releasing a measurable surface charge. A subset of pyroelectrics whose polarisation can be switched by an applied field are the ferroelectrics.
Polar crystals underpin infrared motion sensors (pyroelectric detectors in burglar alarms sense the heat of a moving body), and the ferroelectric members (barium titanate, PZT, lithium niobate) power capacitors, memory, and electro-optic modulators. All 10 polar classes are non-centrosymmetric, so every polar crystal is also piezoelectric, but the reverse fails: many piezoelectrics (like quartz, class 32) are not polar because they have no unique unreversible axis.
Barium titanate below 120 C is class 4mm, polar. Its titanium sits slightly off-centre along the 4-fold axis, giving a permanent dipole that a field can flip: the essence of a ferroelectric.
A polar class has a unique, unreversible axis and a built-in spontaneous polarisation.
Polar (10 classes, pyroelectric) is a strict subset of non-centrosymmetric (21 classes, mostly piezoelectric). Every polar crystal is piezoelectric; not every piezoelectric is polar, quartz is the standard counterexample.