Matplotlib applying polar transformation without polar projection

Question

Is there a way to plot (theta, r) coordinate pairs in the orthogonal 2D space without projecting it to cartesian coordiantes?

That is, I'd like a rectangular plot, where the x-axis is theta, and the y-axis is z, which respects a polar data transformation, such e.g. that data at theta = (2.5)pi is plotted at theta = pi/2. In other words, I need the x-axis to be periodic.

Doing this gives that periodic behavior:

import matplotlib.pyplot as plt
fig = plt.figure(projection='polar')
ax = fig.add_subplot(111)
theta = np.linspace(0, 4*np.pi, 100)
r = theta
plt.show() 

But I do not know how to plot this in the theta, r plane without projecting the polar coordinates into cartesian space.