Symmetric Matrices of Multinormal Distributions

Question

I'm working on some computational mathematics for development in three.js and have been working with out the first parameters of multinormal distributions in Mathematica 12.2. (They're useful in procedural terrain generation because they have nice organic looking properties when mixed together.)

However, after looking pretty deeply at the properties of the covariance matrix such as its eigenvalues, variance, and standard deviation, I found a trait that reduces the randomness of the outputted distributions. They all seem to tilt bottom-left to top-right. (The variance and standard dev. are listed with each plot)

The properties of the covariance matrix used in a multinormal distribution is that it is positive and symmetric. However, I'm wondering what controls the rotation of the spread, and if a top-left, bottom-right orientation is possible? Or is this a fundamental property of positive and symmetric matrix? I thought I knew a lot of matrices but obviously that was wrong, and there is a gap in my knowledge. Investigating on my own it seems fundamental to its nature but I wanted to do a bit more indepth research because my plan of action for the project depends on the root cause of this orientation behaviour.

Updated 2021-01-29

What I was missing was the concept of correlation coefficients from statistics (described minimally in the accepted answer but it is the answer I used in the project). Understanding the relationship between "x", "y", and "p" does return the expected results. Below "p" is a negative exponent.

Answer 1

$${\bf \Sigma} = \begin{pmatrix} \sigma_x^2, \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y, \sigma_y^2 \end{pmatrix}$$

where $\rho \in [-1,1]$.

Answer 2

However, I'm wondering what controls the rotation of the spread, and if a top-left, bottom-right orientation is possible?

Yes, just put minus sign on the off-diagonal terms.

Or is this a fundamental property of positive and symmetric matrix?

The off diagonal terms determine the direction and the shape of your ellipse. As larger it w.r.t. the variance terms (main diagonal), the more "edgy" your ellipse. For positive values you have your current "positive" direction (=correlation), for negative values you'll have the opposite (=negative correlation). For zero terms you will have a perfect circle. All covariance matrices are positive definite. Whatever the orientation of the plot. As long as you don't have degenerate components, that is, with zero variance (main diagonal), your covariance matrix will be positive definite matrix. Otherwise only semi-positive definite matrix.