Count the number of possible drawings Take a square, ABCD. Add two points, E and F on AB such that AE=EF=FB. Now, add G, H on BC, I and J on CD, K and L on AD.
Now pick four pairs of the eight points, E,F,G,H,I,J,K,L. Draw the line segments between your picks, for example EG,FH, IL, JK. If we consider two such drawings equal if they can be rotated into each other, how many different drawings are there? What if we mandate that the drawing must have a line of symmetry?
 A: Start by arranging all the possible pairs in the following form:
\begin{pmatrix}
        \text{EF} & \text{EG} & \text{EH} & \text{EI} & \text{EJ} & \text{EK} & \text{EL} \\
         & \text{FG} & \text{FH}& \text{FI}& \text{FJ}& \text{FK}& \text{FL} \\
        && \text{GH}& \text{GI}& \text{GJ}& \text{GK}& \text{GL} \\
        &&& \text{HI}& \text{HJ}& \text{HK}& \text{HL}\\&&&& \text{IJ}& \text{IK}& \text{IL}\\&&&&& \text{JK}& \text{JL}\\&&&&&& \text{KL}\end{pmatrix}
Now we see that the number of pairing arrangements so that all $4$ pairs contain uncommon elements is $7\times 5\times 3\times 1=105$. Hence there must be $105$ such drawings. Notice that a drawing can be rotated to produce another pair drawing pattern from the other $104$ drawings if it is not symmetric about either the vertical (passing from the midpoints of $LK$ and $GH$) and horizontal (passing from the midpoints of $EF$ and $IJ$) axes, or the diagonal axes (lines $AC$ and $BD$). Also, a drawing can be rotated to produce 3 other drawing pattern from the other $104$ drawings if it is not symmetric about both the vertical (passing from the midpoints of $LK$ and $GH$) and horizontal (passing from the midpoints of $EF$ and $IJ$) axes, and the diagonal axes (lines $AC$ and $BD$).
Let us define diagonalizing as the process of rotating a drawing by $\pm45^\text{o}$ such that each node point is moved one unit forward (e.g. for $+45^\text{o}$ rotation, point $E$ becomes point $L$, $L$ becomes $K$, $K$ becomes $J$ and so on).

Notice that generally $\text{|EF|}\ne\text{|EL|}$, however diagonalizing allows a stretch or shrinkage for compensation.
I didn't know which of the cases you referred to so I listed all of them.

So if diagonalization and mirroring are not allowed then there are $35$ distinct drawings.
If diagonalization is allowed while mirroring is not, then there are $19$ distinct drawings.
If both diagonalization and mirroring are allowed, then there are $17$ distinct drawings.
