Symmetric Mandelbrot Set with Pseudocode

for (int i = 0; i ≤ 500; i++)
{
	for (int j = 0; j ≤ 500; j++)
	{
		x = 0.0;
		y = 0.0;
		xs = -2.5 + (i / 100.0);
		ys = -2.5 + (j / 100.0);
		k = 0;
		do
		{
			k = k + 1;
			xnew = x*x - y*y + xs;
			ynew = 2.0*x*y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ kmax) && (x*x + y*y ≤ 6.25));
		PlotPixel(i, j, color);
	}
}

Asymmetric Mandelbrot Set with Pseudocode

for (int i = 0; i ≤ 500; i++)
{
	for (int j = 0; j ≤ 500; j++)
	{
		x = 0.0;
		y = 0.0;
		xs = -2.5 + (i / 100.0);
		ys = -2.5 + (j / 100.0);
		k = 0;
		do
		{
			k = k + 1;
			xnew = x*x - x*y + xs;
			ynew = 2.0*x*y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ kmax) && (x*x + y*y ≤ 6.25));
		PlotPixel(i, j, color);
	}
}

Mandelbrot	Build: (f(x,y), g(x,y))	Escape: h(x,y)>value	Plot
Symmetric (Basic) Mandelbrot Set	(x² - y², 2.0*x*y)	x² + y² > 6.25	Pixel
Symmetric Mandelbrot Variant #1	(x² - y²-x, 2.0*x*y)	x² + y² > 6.25	Pixel
Symmetric Mandelbrot Variant #2	(x² + x*y², y*x²)	x² + y² > 6.25	Pixel
Symmetric Mandelbrot Variant #3	(x²*y, x²*y)	x² + y² > 6.25	Pixel
Asymmetric Mandelbrot Set	(x² - y*x, 2.0*x*y)	x² + y² > 6.25	Pixel
Asymmetric Mandelbrot Variant #1	(x²-y²-y, 2.0*x*y)	x² + y² > 6.25	Pixel
Asymmetric Mandelbrot Variant #2	(x*y + x²*y, y*x²)	x² + y² > 6.25	Pixel
Asymmetric Mandelbrot Variant #3	(x²*y, x*y²)	x² + y² > 6.25	Pixel

Mandelbrot Variation Sample with Pseudocode and Symmetric Map
Here is a Mandelbrot variation with build (x^mx*y^my, x^nx*y^ny) where mx, my, nx, ny were incremented for each figure.
Exponents ranged from 0 to 4 and yielded 5⁴ or 625 figures.
Each figure's symmetry was examined and a map created showing whether each figure was symmetric or asymmetric.

for (int i = 0; i ≤ 300; i++)
{
	for (int j = 0; j ≤ 300; j++)
	{
		x = 0.0;
		y = 0.0;
		xs = -2.0 + (i / 75.0);
		ys = -2.0 + (j / 75.0);
		k = 0;
		do
		{
			if (mx == 0 && my == 0) xnew = xs;
				else xnew = xmx*ymy + xs;
			if (nx == 0 && ny == 0) ynew = ys;
				else ynew = xnx*yny + ys;
			x = xnew;
			y = ynew;
			k = k + 1;
		} while ((k ≤ kmax) && (x*x + y*y ≤ 4.0));
		PlotPixel(i, j, color);
	}
}

The Symmetric Map shows figure results where:
asymmetric figures are red and symmetric figures are white.
x^mx*y^my is shown as row and x^nx*y^ny is shown as column.

For example, the following yields a symmetric figure:
mx = 3, my = 2, is along the left side as 32.
nx = 4, ny = 3, is along the top as 43.

Real: x³*y²
Imaginary: x⁴*y³

for (int i = 0; i ≤ 500; i++)
{
	for (int j = 0; j ≤ 500; j++)
	{
		xs = 0.39;
		ys = 0.36;
		x = -2.5 + (i / 100.0);
		y = -2.5 + (j / 100.0);
		k = 0;
		do
		{
			k = k + 1;
			xnew = x*x - y*y + xs;
			ynew = 2.0*x*y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ kmax) && (x*x + y*y ≤ 6.25));
		PlotPixel(i, j, color);
	}
}

for (int i = 0; i ≤ 500; i++)
{
	for (int j = 0; j ≤ 500; j++)
	{
		xs = 0.39;
		ys = 0.36;
		x = -2.5 + (i / 100.0);
		y = -2.5 + (j / 100.0);
		k = 0;
		do
		{
			k = k + 1;
			xnew = x*x - y + xs;
			ynew = 2.0*x*y + ys;
			x = xnew;
			y = ynew;
		} while ((k ≤ kmax) && (x*x + y*y ≤ 6.25));
		PlotPixel(i, j, color);
	}
}