Here is a surface you might find on your vacuum cleaner or under the hood of your car:

r = chebfun(@(x) .5+.04*cos(40*x));
F = chebfun2(@(x,phi) 2*x, 'trigy');
G = chebfun2(@(x,phi) r(x).*cos(pi*phi), 'trigy');
H = chebfun2(@(x,phi) r(x).*sin(pi*phi), 'trigy');
surf(F,G,H), axis equal off, camlight

Viewed as a surface, we see that this is periodic in the $\phi$ direction and nonperiodic in the $x$ direction. What's interesting is that its representation as a trio of chebfun2 objects shares this property: each of them is nonperiodic in the first variable and periodic in the second, because the flag 'trigy' has been specified.

Until recently, a chebfun2 representation had to be Chebyshev in both variables or, if 'trig' was specified, trigonometric in both directions. The ability to mix the two with 'trigx' or 'trigy' is new. Here we see some details of the three chebfun2 objects:

F, G, H
F =
   chebfun2 object  (trig in y)
       domain                 rank       corner values
[  -1,   1] x [  -1,   1]        1     [  -2    2   -2    2]
vertical scale =   2 
G =
   chebfun2 object  (trig in y)
       domain                 rank       corner values
[  -1,   1] x [  -1,   1]        1     [-0.47 -0.47 -0.47 -0.47]
vertical scale = 0.54 
H =
   chebfun2 object  (trig in y)
       domain                 rank       corner values
[  -1,   1] x [  -1,   1]        1     [-6.9e-17 -6.9e-17 4.7e-17 4.7e-17]
vertical scale = 0.53 

The F chebfun2 is trivial, but G is interesting. The command plotcoeffs shows how it mixes different representations in the two directions. The H chebfun2 looks much the same.

plotcoeffs(G)

One reason for representing a periodic function periodically is that it is somewhat more efficient. More important in practice is that its derivatives may retain their smoothness and accuracy at the wraparound point. Perhaps a reader of this example will have an idea of how to extend it to illustrate the advantages of 'trigy'?