Hello Marie

If you are not concerned about the overlap you could convert the blocks to centroids then do the spatial join…

if you are concerned about the overlap (percentage the block falls into the overlaying polygon) then you could use the intersect command which may assign a block to two adjacent polygons…

Below is something I found on the ESRI resources website…it may provide you with the use case that you need. 

Select polygon using polygon

Select polygon using polygon
INTERSECT	A, C, D, E, F, G, H, I, J, K, M
CONTAINS	C, E, H, M
COMPLETELY_CONTAINS	C
CONTAINS_CLEMENTINI	C, E, H, M
WITHIN	F, G, H, M
COMPLETELY_WITHIN	F
WITHIN_CLEMENTINI	F, G, H, M
ARE_IDENTICAL_TO	H, M
BOUNDARY_TOUCHES	D, E, G, H, I, J, M
SHARE_A_LINE_SEGMENT_WITH	D, H, I, M
CROSSED_BY_THE_OUTLINE_OF	A, E, G, J, K
HAVE_THEIR_CENTER_IN	C, E, F, G, H, K, L

From: ctpp-news-bounces@chrispy.net [mailto:ctpp-news-bounces@chrispy.net] On Behalf Of Marie Bousfield
Sent: Friday, August 22, 2014 10:54 AM
To: ctpp-news@chrispy.net
Subject: [CTPP] spatial join

In Arcview 10, I am trying to join the 2010 census blocks to another polygon feature with city districts.  I am using the spatial join (within).  The problem is that the blocks at the borders of the districts are not joined. 

Any help?

Is there a better way??

Thanks!!   

Marie Bousfield