Yesterday I read in some texts on theta characteristics of algebraic curves, which are structures on their Jacobians... Do you know if someone has thought if such, but more general, things can exist on motives?
The existence of theta characteristics on a curve is a refinement of the splitting of its motive. The reference to this is Oliver Ronding's theta characteristics and stable homotopy type paper. The point is that the existence of a theta characteristic on a curve (with a rational point) implies a splitting in the motivic stable homotopy category $SH$: $X \simeq S^{0,0} \vee J(X) \vee S^{2,1}$
This realizes to the splitting in category of motives, $M(X) \simeq \mathbb{Z} \oplus Jac(X) \oplus \mathbb{Z}(1)[2]$ (which always happens since the motive of an algebraic variety of dimension $d$ with a rational point splits into $\mathbb{Z} \oplus Pic(X) \oplus ?? \oplus Alb(X) \oplus \mathbb{Z}(d)[2d]$). But the splitting in the $SH$ is much finer  for example it implies splitting as a module over the motivic Steenrod algebra.
My guess is that a theta characteristic on a motive should be a structure on its stable motivic homotopy type, as opposed to its motive and it should lead to a splitting of the top cell. Hints of this is given in the Rondings paper: he shows that all you need for this refined splitting is that the SpanierWhitehead dual of $X$ is of a certain shape (Corollary 5.2).

1$\begingroup$ Pic(X) instead of Jac for a general X? $\endgroup$ – Mikhail Bondarko Jul 20 '16 at 21:55

$\begingroup$ Thanks to Andrew Ranicki, Roendigs' article is fortunately online: maths.ed.ac.uk/~aar/papers/roendigs.pdf , and a "thanks!" for that interesting hint! Actually, I came along that question through a remark by Jack Morava (he stressed that Freed's arxiv.org/abs/hepth/0607134 is very interesting, making one read in arxiv.org/abs/math/0211216 ) $\endgroup$ – Thomas Riepe Jul 21 '16 at 10:19

$\begingroup$ wow  very cool. I should also say that motivic homotopy theory (as opposed to the theory of motives) is particularly good at detecting "quadratic" information (I can be more precise, but I fear that I will end up spamming); maybe the motivic connection might really be in these "Riemann parity" $\endgroup$ – Elden Elmanto Jul 21 '16 at 19:50