What symbols, preferably secular, that signify the concept of
non-judgemental understanding
would be appropriate for a tattoo?

This was a fascinating question to research! Below are links to images
of some symbols for you to consider.

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Here are some West African adinkra symbols that may interest you:

 AKOMA NTOSO
"linked hearts" 
 symbol of understanding and agreement

http://www.welltempered.net/adinkra/htmls/symbols/akon.htm

 BI NKA BI
"No one should bite the other" 
 symbol of peace and harmony

WellTempered.net: Adinkra Symbols & Meanings
http://www.welltempered.net/adinkra/htmls/adinkra/bink.htm

 MPATAPO
"knot of pacification/reconciliation" 
 symbol of reconciliation, peacemaking and pacification

WellTempered.net: Adinkra Symbols & Meanings
http://www.welltempered.net/adinkra/htmls/adinkra/mpat.htm

 Osram. (the moon). 
 Symbol of faith, patience, understanding and determination. 

Camel's Closet: Symbology
http://www.camelscloset.com/symbology.htm

Here you will find an index of other adinkra symbols and their
meanings:

WellTempered.net: Adinkra Symbols & Meanings
http://www.welltempered.net/adinkra/htmls/adinkra_index.htm

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Among the meanings of the astrological symbol for Jupiter is this one:

"Jupiter... also stands for empathy and philanthropy. When used in
psychological astrology Jupiter indicates the ability to understand
and adapt to the established social structure and the desire to expand
in knowledge, for instance, via studies and travelling."

Symbols.com
http://www.symbols.com/encyclopedia/17/171.html

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The Blissymbol for "empathy" represents the union of emotion and
knowledge:

Symbols.net
http://www.symbols.net/blissymbolics/feeling.html

The "building-blocks" of Blissymbolics:

Symbols.net
http://www.symbols.net/blissymbolics/dictionary.html

More on Blissymbolics:

Blissymbols.org
http://www.blissymbols.org/

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Here is the Kanji character symbolizing "understanding":

Built to Last: Kanji Directory
http://www.abuilttolast.com/kanji/understanding.html

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In the Kabbalah, the Sephirah Binah symbolizes understanding:

Argotique
http://www.argotique.com/qabalah/binah.htm

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The heart chakra represents empathy and forgiveness:

The Heart Chakra lies at the centre of the seven main chakras and
marks a transition point between the personal chakras below and the
universal chakras above.

It is a powerful centre of transformation.

When we enter the heart space we begin to connect with people on a
deeper level. We are able to experience understanding, empathy,
forgiveness, love and compassion. We become kinder to ourselves as
well as others.

Shola Arewa 
http://www.shola.co.uk/workshop/chakra3.php

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You might want to consider using blue as the primary color of your
tattoo:

Blue symbolizes: understanding, harmony, sea & sky (all-encompassing),
peace, sincerity, faith, fidelity, truth, spirituality, culture,
philosophy, calmness (light blue), healing, love, nobility, protection
against evil, longing, eternity, innocence, heaven, loyalty (true
blue), and public servant.

Hate Free Zone
http://www.hatefreezone.org/index.asp?name=siteCredit

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My search strategy in locating this information consisted of several
Google searches, plus an extensive search through the indexes of
Symbols.com and Symbols.net, two of the most valuable online sources
for information about graphic symbols.

Symbols.com
http://www.symbols.com

Symbols.net
http://www.symbols.net

Search terms used:

"symbol"
"symbology"
"ideogram"
"pictogram"
"motif"
"understanding"
"acceptance"
"empathy"
"adinkra"
"kanji"
"kabbalah"

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I hope this will be useful. If anything is unclear, or if a link
doesn't work for you, please request clarification; I'll gladly offer
further assistance before you rate my answer.

Best wishes,
pinkfreud

Thorough and helpful.  Thanks much.