Question: Say I have 2 continuous variables (x & y) in a logit model. I want to test the following hypotheses in the model:

1. For low values of x, increasing the value of y will increase the value of the response variable
2. For high values of x, increasing the value of y will decrease the value of the response variable

Is there a way to specify this using interaction terms? I have only used interaction terms in which y has a linear impact on the effect of x on the response variable.

Answer: 
It seems to me that you want:

1. A two-way interaction, $XZ$, between $X$ and $Z$. This will estimate the expected change in the response for a 1 unit change in $X$, when $Z$ also changes by 1 unit.
2. If you want this two-way interaction also to also vary with $X$ then you can interact the two-way interaction with $X$, which will be $X^2Z$. This 3-way interaction will estimate the expected change in the response for a 1 unit change in $X$ when the interaction $XZ$ also changes by 1 unit.
X1 = I(x>a)

X2= I (x<b)

Logit (Response ) = beta0 + betax \* x + betay \* x + betax1y \* X1 \* y + betax2y \* X2 \* y

Then X1*y is activated when x is above a, X2* y when x is below b.
You can exclude betay*y if you think that response is only through y*x terms.
A and b are exogenous parameters in this model, so either outside knowledge or additional testing on which fit best is needed.