Quantum Mechanics, Particle in a box

[geocentric]

Consider a particle in a 1-D box with impenetrable walls at x=0 and x=L. Then the eigen function for this problem is of the form,
psi(x)= sqrt(2/L)sin(n(pi)x/L) ------> (1)
This is always of odd parity. The question which is troubling me is that, in this case though the eigen function seems to be of odd parity mathematically, when see their eigen states, they are alternately symmetric and anti-symmetric i.e odd and even parity.
On the contrary, if the origin is shifted to the centre of the box in which case the particle is confined between -L/2 to +L/2, then there are 2 sets of eigen functions. One set of functions is of odd parity and other set of functions is of even parity. Since both are of same width, how can there exist such a difference? Moreover, in the first case, why doesn't the eigen function{i.e eq'u (1)} not conform with the eigen state representation diagram? Could someone point out the flaw in my understanding?

[Enthalpy]

Could it be just a matter of vocabulary? Like "parity" having one meaning for mathematical function analysis and another in QM, like "what happens if I swap two particles" or something similar?

What you call parity in the sine functions reminds me what's called symmetry in my mothertonge, and this symmetry changes when one displaces the origin, yes.