Suppose that $\nabla$ is a linear connection on a vector bundle $E\to M$, and that there is $\sigma\in \Omega^1(M;E)$, a 1-form on $M$ with values in $E$ such that $\sigma_x:T_xM\to E$ E_x$is a linear isomorphism. This is called a soldering form. It identifies$E$with$TM$. The torsion is then$d^{\nabla}\sigma\in\Omega^2(M;E)$. It is an obstruction against the soldering form being parallel for$\nabla$. Maybe this explains, that space is twisting along geodesics if the torsion is non-zero. So torsion can be viewed either as a property of the soldering form (choose it better if you want to get rid of torsion), or as a property of$\nabla$(if you identify$TM$with$E$with the given soldering form). This works also with$G$-structures on$M$. Consider a principal$G$-bundle$P\to M$and a representation$\rho:G\to GL(V)$where$\dim(V)=\dim(M)$. A soldering form is now a$G$-equivariant and horizontal 1-form$\sigma\in\Omega^1(P,V)^G_{hor}$which is fiberwise surjective. This induces a form$\bar\sigma\in\Omega^1(M,P\times_G V)$which is a soldering form in the sense above. You can compute torsion either on$P$or on$M$and they correspond to each other. This ties in with the answer of Chris Schommer-Pries. 1 Let me expand a little the answer of José Figueroa-O'Farrill. Suppose that$\nabla$is a linear connection on a vector bundle$E\to M$, and that there is$\sigma\in \Omega^1(M;E)$, a 1-form on$M$with values in$E$such that$\sigma_x:T_xM\to E$is a linear isomorphism. This is called a soldering form. It identifies$E$with$TM$. The torsion is then$d^{\nabla}\sigma\in\Omega^2(M;E)$. It is an obstruction against the soldering form being parallel for$\nabla$. Maybe this explains, that space is twisting along geodesics if the torsion is non-zero. So torsion can be viewed either as a property of the soldering form (choose it better if you want to get rid of torsion), or as a property of$\nabla$(if you identify$TM$with$E$with the given soldering form). This works also with$G$-structures on$M$. Consider a principal$G$-bundle$P\to M$and a representation$\rho:G\to GL(V)$where$\dim(V)=\dim(M)$. A soldering form is now a$G$-equivariant and horizontal 1-form$\sigma\in\Omega^1(P,V)^G_{hor}$which is fiberwise surjective. This induces a form$\bar\sigma\in\Omega^1(M,P\times_G V)$which is a soldering form in the sense above. You can compute torsion either on$P$or on$M\$ and they correspond to each other. This ties in with the answer of Chris Schommer-Pries.