Given a Euclidean Jordan algebra (V,o,〈.,.〉) with the (corresponding) symmetric cone K, a linear transformation L:V→V and qεV, the linear complementarity problem LCP(L,q) is to find a vector xεV such thatxεK, y:=L(x)+qεK and xy=0. To investigate the global uniqueness of solutions in the setting of Euclidean Jordan algebras, the P-property and its variants of a linear transformation were introduced in Gowda et al. (2004) [3] and it is shown that if LCP(L,q) has a unique solution for all qεV, then L has the P-property but the converse is not true in general. In the present paper, when (V,o,〈,〉) is the Jordan spin algebra, we show that LCP(L,q) has a unique solution for all qεV if and only if L has the P-property and L is positive semidefinite on the boundary of K. © 2015 Elsevier Inc. Allrightsreserved.