Let $\triangle ABC$ be inscribed in circle $O$ with $\angle ABC = 36^\circ$. $D$ and $E$ are on the circle such that $\overline{AD}$ and $\overline{CE}$ are diameters of circle $O$. List all possible positive values of $\angle DBE$ in degrees in order from least to greatest.

[i]Proposed by Ambrose Yang[/i]
1. **Identify the given information and draw the diagram:**
   - $\triangle ABC$ is inscribed in circle $O$.
   - $\angle ABC = 36^\circ$.
   - $D$ and $E$ are points on the circle such that $\overline{AD}$ and $\overline{CE}$ are diameters of circle $O$.

2. **Determine the implications of $AD$ and $CE$ being diameters:**
   - Since $AD$ and $CE$ are diameters, $\angle AOD = 180^\circ$ and $\angle COE = 180^\circ$.
   - This implies that $A$, $D$, $C$, and $E$ are all points on the circle, and $AD$ and $CE$ are straight lines through the center $O$.

3. **Analyze the position of points $B$ and $C$ relative to $AD$:**
   - Since $B$ and $C$ are on opposite sides of $AD$, $\angle CBD = 180^\circ - \angle ABC = 180^\circ - 36^\circ = 144^\circ$.
   - This means $\angle CBD = 144^\circ$.

4. **Analyze the position of points $A$ and $B$ relative to $CE$:**
   - If $A$ and $B$ are on opposite sides of $CE$, then $\angle ABE = 180^\circ - \angle ABC = 180^\circ - 36^\circ = 144^\circ$.
   - Thus, $\angle DBE = \angle CBD + \angle ABE = 144^\circ + 36^\circ = 180^\circ$.

5. **Consider the case where $A$ and $B$ are on the same side of $CE$:**
   - If $A$ and $B$ are on the same side of $CE$, then $\angle DBE = \angle ABC = 36^\circ$.

6. **List all possible positive values of $\angle DBE$:**
   - The possible values of $\angle DBE$ are $36^\circ$ and $144^\circ$.

The final answer is $\boxed{36^\circ, 144^\circ}$.