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As observed, the quadratic term may be equivalently removed from the inequality due to different homogeneity; then $x^*a^*ax+a^*x^*xa\leq a^*x^*ax+x^*a^*xa$ can be rewritten $[a,x]^*[a,x]\le0$, so the condition is exactly: commutativity of $C^*$$A$.
As observed, the quadratic term may be equivalently removed from the inequality due to different homogeneity; then $x^*a^*ax+a^*x^*xa\leq a^*x^*ax+x^*a^*xa$ can be rewritten $[a,x]^*[a,x]\le0$, so the condition is exactly: commutativity of $C^*$.
As observed, the quadratic term may be equivalently removed from the inequality due to different homogeneity; then $x^*a^*ax+a^*x^*xa\leq a^*x^*ax+x^*a^*xa$ can be rewritten $[a,x]^*[a,x]\le0$, so the condition is exactly: commutativity of $A$.
As observed, the quadratic term may be equivalently removed from the inequality due to different homogeneity; then $x^*a^*ax+a^*x^*xa\leq a^*x^*ax+x^*a^*xa$ can be rewritten $[a,x]^*[a,x]\le0$, so the condition is exactly: commutativity of $C^*$.
