Are there statistical techniques that investigate such relationships ...? If we have data set, X and Y variable. Say, we do correlation analysis and get some correlation coefficient. Besides, we find an important fact after observing their relationship: That is, the scatter plot of X and Y has a triangular shape. Which means that (for example) when X values are increasing Y values are increasing for all X values, whereas vice versa is not true; when the Y value is increasing X values are anything for all Y values.
What kind of analysis should I do to investigate this?  

(Update in response to @Penguin_Knight)
Your graph is exactly what I explained. Just take away the negative values with an imaginary y=0 line. As you can see there are many data points the X of which is either 0 or very small, and Y has pretty high value. However there is no data point that have y=0 and x is quite high value. And as you can see it makes the data scatter shape a right angle triangle. Thus we could say that the high X values necessitates high Y values but high Y values does not always have high X values. I find it very interesting. In practice for example I measure Complexity and Size of given entities. And my analysis show that all complex entities have big size but not all big-size entities are complex. Then I conclude that the certain amount of complexity requires defined amount of size. You cannot put more complexity in a given size. This is a bit abstract but you see my point? It is very interesting and I would like to get some help on how this kind of relationships are discussed or described in statistics formally.
 A: Sorry, can't wrap my head around it... 
From the plot above (the red line is when x = y)... I can see your first condition where when x increases, y can only increase. But I don't understand given the first condition, how can "X values are anything for all Y values."
Please post your scatter plot, while I'll go prepare some popcorn.
A: Typically, to say a scatterplot has triangular shape, it would have to look like this: /\. What you are describing sounds like this: /.
The part about Y~X being positively correlated in the plot, and X~Y not, is IMPOSSIBLE. Please check your code, or the description. 
In general you can use a regression model to model a relationship between a ratio scaled X and Y. Look up the assumptions for different regression models in a book or on the Wiki.
A: The question is rather vague. From your description, the only case where I can imagine the scatterplot to look like a triangle (and fit the description), is a case where the variance of the Y increases with the X values. The scatter plot would appear something like:
|           .
|         ...  
|       ..... 
|    ........
|  ......     
|.... 
+----------------

What is the aim of your analysis? If you're trying to predict the Y from the X, you might want to take the increasing variance into account during the analysis using something like generalized least squares or other such methods. If you want to find other such cases from a larger data set that contains several X variables, as crude solution you might consider calculating, e.g., a range or variance of the Y variable along the X variable. Generating all the possible plots, and eyeballing them for interesting patterns might be a viable option, also.
A: It sounds like the lower bound on y is a function of x. Something like this? 

