Chirality of buta-1,2-diene
Let's see if a picture can help.  Newman projections can be handy in analyzing the stereochemistry of allenes.  On the left is a Newman projection of methyl allene (buta-1,2-diene).  There is a sigma plane of symmetry that is perpendicular to the screen and contains the 3 allenic carbons plus the methyl group.  Any molecule with a plane of symmetry is achiral.  On the other hand, look at the Newman projection to the right, 1,3-dimethyallene (penta-2,3-diene).  There is no plane symmetry in this case, only a $\ce{C_2}$ axis.  Draw the mirror image of the pictured allene and you will find that it is not superimposable on its mirror image. 1,3-dimethylallene is a chiral molecule.