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The Jordan algebra of the symmetric matrices of order two over a field $K$has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. Wedescribe the graded polynomial identities for these two gradings when the basefield is infinite and of characteristic different from 2. We exhibit bases forthese identities in each of the two cases. In one of the cases we perform aseries of computations in order to reduce the problem to dealing withassociators while in the other case one employs methods and results fromInvariant theory. Moreover we extend the latter grading to a$\mathbb{Z}_2$-grading on $B_n$, the Jordan algebra of a symmetric bilinearform in a vector space of dimension $n$ ($n=1$, 2, \dots, $\infty$). We callthis grading the \textsl{scalar} one since its even part consists only of thescalars. As a by-product we obtain finite bases of the $\mathbb{Z}_2$-gradedidentities for $B_n$. In fact the last result describes the weak Jordanpolynomial identities for the pair $(B_n, V_n)$.

2-graded polynomial identities for the Jordan algebra of the symmetric matrices of order two