Pretty sure this has to do with the coordinate system xna used i think all the matrices are right handed in xna. Yep i would definately ditch the unity system even if i had to write a new set of full projects that are monogame versions of my old ones the reason is Math.
If you really are intent on staying with the unity coordinates.
Things i would try before i refactored the entire project would be.
in 3d try
setting your eye matrix / your view camera's up to down. i.e. your vector3 up to *= -1;
xna ms matrix page
multiplying your world matrixs before passing your effect draws by a matrix identity
that has its .M22 row colum set to = -1 instead of the default 1
e.g . m11 = 1 m22 = -1 m33 = 1
if you try to do by the world though that might mess things up.
You could try setting m24 to -1 instead.that is essentially the skewing portion w of the matrix.
you will probably need to demo test that first.
If its 2d, you could invert all your values around y screenHeight sort of a pain in the ass to do that and confusing it would mean a performance hit as well but i guess it would work.
wrap a spriteBatch call and use that.
rectangle n.Y = -(r.Y - screenHeight) + screenHeight;
ex rectangle 0,0,1,1 screen width height 1080, 720 the Y value becomes 0, (0-1) + 720 , 1080,720 then calling spritebatch (,,,,, SpriteEffects.FlipVertically); placing the rectangle at 719 drawn to 720 from down to up.
This and any of these solutions will make it tough to use the math class and a lot of math functions. That is the main problem with flipping the y to be like unitys. Later it will cause you headache after headache
Personally if i was you if your switching from unity to monogame i would do it all the way and commit if your gonna do it, do it. Think of it as a break to go thru all your code and review it.
Get your head into top down not down up. To be honest down up is a relic from the old days, when the coordinate system used in early gdi's, aligned to a drawing graph with a absolute coordinate system in the +,+ region. Its far less intuitive in practice. Internally mono game is relying on gl and dx though gl is left handed and dx is right handed. monogame sets up the conversion somewere for you but that deals with the x axis primarily and the order of matrix arthmitic.