In classical mechanics, change in momentum $\Delta \mathbf p$ and change in kinetic energy $\Delta T$ of a particle are defined as follows in terms of the net force acting on the particle $\mathbf F_\text{net}$, where in each case the integrations are done over the path taken by the particle through spacetime. $$\begin{align} \Delta T &= \int \mathbf F_\text{net} \cdot d\mathbf x \\ \Delta \mathbf p &= \int \mathbf F_\text{net} dt \end{align}$$

This suggests some sort of correspondence.

$$\begin{align} \mathbf x &\longleftrightarrow T \\ t & \longleftrightarrow \mathbf p \end{align}$$

Noether's theorem provides an association between physical symmetries and conserved quantities.

$$\begin{align} \text{symmetry in time} &\longleftrightarrow \text{conservation of energy} \\ \text{symmetry in position} &\longleftrightarrow \text{conservation of momentum} \end{align}$$

Additionally, when studying special relativity, there is a similar suggested correspondence between the components of the position four-vector $\mathbf X$ and the energy-momentum four-vector $\mathbf P$. Here, $E$ represents total energy $E = mc^2 + T + \mathcal O \left( v^3/c^3 \right)$

$$\begin{array} \ \mathbf X = \begin{bmatrix} ct \\ x \\ y \\ z \end{bmatrix} & \mathbf P = \begin{bmatrix} E/c \\ p_x \\ p_y \\ p_z \end{bmatrix} \end{array}$$ Thus, comparing components, and discarding factors of $c$ the following correspondence is suggested. $$ \begin{align} t &\longleftrightarrow E \\ \mathbf x &\longleftrightarrow \mathbf p \end{align} $$

Is there an underlying relationship between these correspondences I have pointed out? And, if so, why are the ones for the classical definitions of kinetic energy and momentum swapped compared to the ones arising from Noether's theorem and special relativity?

In classical mechanics, change in momentum $\Delta \mathbf p$ and change in kinetic energy $\Delta T$ of a particle are defined as follows in terms of the net force acting on the particle $\mathbf F_\text{net}$, where in each case the integrations are done over the path taken by the particle through spacetime. $$\begin{align} \Delta T &= \int \mathbf F_\text{net} \cdot d\mathbf x \\ \Delta \mathbf p &= \int \mathbf F_\text{net} dt \end{align}\tag{1}$$

This suggests some sort of correspondence.

$$\begin{align} \mathbf x &\longleftrightarrow T \\ t & \longleftrightarrow \mathbf p \end{align}\tag{2}$$

Noether's theorem provides an association between physical symmetries and conserved quantities.

$$\begin{align} \text{symmetry in time} &\longleftrightarrow \text{conservation of energy} \\ \text{symmetry in position} &\longleftrightarrow \text{conservation of momentum} \end{align}\tag{3}$$

Additionally, when studying special relativity, there is a similar suggested correspondence between the components of the position four-vector $\mathbf X$ and the energy-momentum four-vector $\mathbf P$. Here, $E$ represents total energy $E = mc^2 + T + \mathcal O \left( v^3/c^3 \right)$

$$\begin{array} \ \mathbf X = \begin{bmatrix} ct \\ x \\ y \\ z \end{bmatrix} & \mathbf P = \begin{bmatrix} E/c \\ p_x \\ p_y \\ p_z \end{bmatrix} \end{array}\tag{4}$$ Thus, comparing components, and discarding factors of $c$ the following correspondence is suggested. $$ \begin{align} t &\longleftrightarrow E \\ \mathbf x &\longleftrightarrow \mathbf p \end{align}\tag{5} $$

Is there an underlying relationship between these correspondences I have pointed out? And, if so, why are the ones for the classical definitions of kinetic energy and momentum swapped compared to the ones arising from Noether's theorem and special relativity?