Counting polydominoes
Ruby, 266 250 242 bytes

-&gt;n,*z{j=(h=1,m=1i,-1,*r=[-m,0]).product h;(r.map{|c|j.map{|v,w|k=r+[v+=c,v+w];k|k==k&amp;&amp;(w=j.map{|y,q|(x=k.map{|v|y*[v,v.conj][q*q]}).map{|g|g-x.map(&amp;:real).min-m*x.map(&amp;:imag).min}.sort_by &amp;:rect})-z==w&amp;&amp;z&lt;&lt;w[0]}};r,*z=z)until r[-n];z.size+1}

Try it online!
Not particularly original, I just recycled my solution to a similar problem, and changed some small details:

start with the basic domino (instead of single tile)
add 2 tiles on each step (instead of one)
check for reflection by using the complex conjugate (instead of only rotation using powers of i)
stop when reaching size 2n (instead of n) -&gt; Accept 2n as input

How it works:
See the linked post for the full explanation. Basically, use complex numbers to make it easier.
Performance:
43 seconds for n=5 (size 10) on TIO
Thanks Dingus for saving some bytes, the function is a mess and could probably be much shorter anyway if implemented properly.