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Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a symmetric tensor provided that $\nabla$ was torsion-free, which Robert pointed out but I missed in the first moment.

(This is because $$\nabla^2f[Y, X] = \partial_X\partial_Y f + \partial_{[Y, X]} f - \partial_{\nabla_Y, X} = \partial_X\partial_Y f - \partial_{\nabla_X, Y}+ T(X, Y),$$ where $T$ is the torsion tensor of nabla.)

Now given such a torsion-free connection, you can associate to an element polynomial $p \in TM \odot TM$ the operator $$Pf = \langle p, \nabla^2 f\rangle.$$ This gives a splitting of your sequence, since the principal symbol of $P$ will be $p$ again.

Conversely, if $S: TM \odot TM \longrightarrow \tau M$ is such a splitting, set $\Gamma$ to be the projection onto $TM$ along the image of $S$ (i.e. $\Gamma = \iota^{-1}(\mathrm{id} - \pi \circ S)$ with $\pi:\tau M \longrightarrow TM \odot TM$ the projection). Then $$\nabla_X Y := \Gamma(\partial_X \partial_Y)$$ is a torsion-free connection.

Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a symmetric tensor provided that $\nabla$ was torsion-free, which Robert pointed out but I missed in the first moment.

(This is because $$\nabla^2f[Y, X] = \partial_X\partial_Y f + \partial_{[Y, X]} f - \partial_{\nabla_Y, X} = \partial_X\partial_Y f - \partial_{\nabla_X, Y}+ T(X, Y),$$ where $T$ is the torsion tensor of nabla.)

Now given such a torsion-free connection, you can associate to an element polynomial $p \in TM \odot TM$ the operator $$Pf = \langle p, \nabla^2 f\rangle.$$ This gives a splitting of your sequence, since the principal symbol of $P$ will be $p$ again.

Conversely, if $S: TM \odot TM \longrightarrow \tau M$ is such a splitting, set $\Gamma$ to be the projection onto $TM$ along the image of $S$ (i.e. $\Gamma = \iota^{-1}(\mathrm{id} - S \circ \pi)$ with $\pi:\tau M \longrightarrow TM \odot TM$ the projection). Then $$\nabla_X Y := \Gamma(\partial_X \partial_Y)$$ is a torsion-free connection.

Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a symmetric tensor provided that $\nabla$ was torsion-free, which Robert pointed out but I missed in the first moment.

(This is because $$\nabla^2f[Y, X] = \partial_X\partial_Y f + \partial_{[Y, X]} f - \partial_{\nabla_Y, X} = \partial_X\partial_Y f - \partial_{\nabla_X, Y}+ T(X, Y),$$ where $T$ is the torsion tensor of nabla.)

Now given such a torsion-free connection, you can associate to an element polynomial $p \in TM \odot TM$ the operator $$Pf = \langle p, \nabla^2 f\rangle.$$ This gives a splitting of your sequence, since the principal symbol of $P$ will be $p$ again.

Conversely, if $S: TM \odot TM \longrightarrow \tau M$ is such a splitting, set $\Gamma$ to be the projection onto $TM$ along the image of $S$ (i.e. $\Gamma = \mathrm{id} - \pi \circ S$ with $\pi:\tau M \longrightarrow TM \odot TM$ the projection). Then $$\nabla_X Y := \Gamma(\partial_X \partial_Y)$$ is a torsion-free connection.

Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a symmetric tensor provided that $\nabla$ was torsion-free, which Robert pointed out but I missed in the first moment.

(This is because $$\nabla^2f[Y, X] = \partial_X\partial_Y f + \partial_{[Y, X]} f - \partial_{\nabla_Y, X} = \partial_X\partial_Y f - \partial_{\nabla_X, Y}+ T(X, Y),$$ where $T$ is the torsion tensor of nabla.)

Now given such a torsion-free connection, you can associate to an element polynomial $p \in TM \odot TM$ the operator $$Pf = \langle p, \nabla^2 f\rangle.$$ This gives a splitting of your sequence, since the principal symbol of $P$ will be $p$ again.

Conversely, if $S: TM \odot TM \longrightarrow \tau M$ is such a splitting, set $\Gamma$ to be the projection onto $TM$ along the image of $S$ (i.e. $\Gamma = \iota^{-1}(\mathrm{id} - \pi \circ S)$ with $\pi:\tau M \longrightarrow TM \odot TM$ the projection). Then $$\nabla_X Y := \Gamma(\partial_X \partial_Y)$$ is a torsion-free connection.

Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a symmetric tensor provided that $\nabla$ was torsion-free, which Robert pointed out but I missed in the first moment.

(This is because $$\nabla^2f[Y, X] = \partial_X\partial_Y f + \partial_{[Y, X]} f - \partial_{\nabla_Y, X} = \partial_X\partial_Y f - \partial_{\nabla_X, Y}+ T(X, Y),$$ where $T$ is the torsion tensor of nabla.)

Now given such a torsion-free connection, you can associate to an element polynomial $(X, p) \in TM \oplus (TM \odot TM)$ the operator $$Pf = \langle p, \nabla^2 f\rangle+ \langle X, d f\rangle.$$ Clearly, this is an element in $\tau M$. Conversely, given $P \in \tau M$, let $p$ be its principal symbol. Then $X:= P - \langle p, \nabla^2\rangle$ is a vector field. Hence the map that maps $(p, X)$ to $P$ is an isomorphism from $\tau M$ to $TM \oplus (TM \odot TM)$.

Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a symmetric tensor provided that $\nabla$ was torsion-free, which Robert pointed out but I missed in the first moment.

(This is because $$\nabla^2f[Y, X] = \partial_X\partial_Y f + \partial_{[Y, X]} f - \partial_{\nabla_Y, X} = \partial_X\partial_Y f - \partial_{\nabla_X, Y}+ T(X, Y),$$ where $T$ is the torsion tensor of nabla.)

Now given such a torsion-free connection, you can associate to an element polynomial $p \in TM \odot TM$ the operator $$Pf = \langle p, \nabla^2 f\rangle.$$ This gives a splitting of your sequence, since the principal symbol of $P$ will be $p$ again. 

Conversely, if $S: TM \odot TM \longrightarrow \tau M$ is such a splitting, set $\Gamma$ to be the projection onto $TM$ along the image of $S$ (i.e. $\Gamma = \mathrm{id} - \pi \circ S$ with $\pi:\tau M \longrightarrow TM \odot TM$ the projection). Then $$\nabla_X Y := \Gamma(\partial_X \partial_Y)$$ is a torsion-free connection.