What, you expect me to do this by hand?!?

make_my_matrix_symmetric.py

matrix_str = """\\sigma^2_{\\text{intercept}} &                       &                             &                               &                                          &               
		\\rho_{\\text{intercept} x} & \\sigma^2_x & \\ldots
		\\rho_{\\text{intercept} x^2} & \\rho_{x x^2} & \\sigma^2_{x^2} & \\ldots 
		\\rho_{\\text{intercept} x^3} & \\rho_{x x^3} & \\rho_{x^2 x^3} & \\sigma^2_{x^3} & \\ldots 
		0                           & \\rho_{x (x - \\kappa_1)^3_+} & \\rho_{x^2 (x-\\kappa_1)_+^3} & \\rho_{x^3 (x - \\kappa_1)_+^3} & \\sigma^2_{(x - \\kappa_1)_+^3}            & \\ldots                        
		0                           & 0                           & \\rho_{x^2 (x-\\kappa_2)_+^3} & \\rho_{x^3 (x-\\kappa_2)_+^3}   & \\rho_{(x-\\kappa_1)_+^3 (x-\\kappa_2)_+^3} & \\sigma^2_{(x - \\kappa_2)_+^3} 
		0 & 0 & 0 & \\rho_{x^3 (x - \\kappa_3)_+^3} & \\rho_{(x - \\kappa_1)_+^3 (x - \\kappa_3)_+^3} & \\rho_{(x - \\kappa_2)_+^3 (x - \\kappa_3)_+^3} & \\sigma^2_{(x - \\kappa_3)_+^3} """

lines = matrix_str.split("\n")
matrix = [line.split("&") for line in lines]
matrix = [[el.strip() for el in line] for line in matrix]

matrix_out = [["" for x in range(7)] for y in range(7)]
for i in range(7):
    for j in range(i + 1):
        matrix_out[i][j] = matrix[i][j]
        matrix_out[j][i] = matrix[i][j]

for i in range(7):
    for j in range(7):
        print(matrix_out[i][j], end="")
        if j < 6:
            print("&", end="")
        else:
            print("\\\\")