Are Terms of Trade reciprocal? I am trying to learn the meaning of Terms of Trade, but I'm confused.
From the wikipedia article, it says that TOT is the ratio of the value of exports over imports,
and since for the other country these are reversed the TOT of the other country is the inverse,
but I don't understand how that works.

Suppose country X with currency symbol \$ trades with country Y with currency symbol ¥.
(I am using an exchange rate of \$1 = ¥5, but it is not relevant.)

We consider two commodities, A and B.  Companies buy inputs and sell output either buying wholesale and selling retail,
or buying inputs to industry and selling the product of that industry.

Input A can be bought in Y at ¥10.  Since in X it costs \$2.20 to buy it, company XA buys 3 million units from the market in Y.
Conversely B can be bought in X at $2, but it costs ¥12 in Y, so company YB buys 3 million units from the market in X.

Since country X exports 3M units of A, which in X are valued at \$2.20, the value of its exports is \$6.6M.
Since it imports 3M units of B valued at \$2, the value of its imports is $6M and 1.1 is its TOT.

Country Y exports 3M units of B valued at ¥12, imports 3M units of A valued at ¥10 for 1.2 as its TOT.

I don't understand why TOT should be reciprocal when they are 1.1 and 1.2.

EDIT: Maybe I am doing it wrong.  What is the meaning of "value" as in the value of exports and value of imports.  I assumed it is the cost if someone bought it locally.
Terms of trade is a measure of relative prices, and prices only. There are no quantities involved. As a recent review paper puts it:

"with exceptions that I will note below, most trade economists have defined a country’s terms of trade as the price of its exports divided by the price of 
its imports, $p_X/p_M$" (p.2). 

Therefore, by definition, in a two country model, they are reciprocal. In a model with more countries, there can be differences, as some countries could import or export different goods. 

Actually, I don't know which Wikipedia you are looking at, but in the English version, the first line states:

"Terms of trade (TOT) refers to the relative price of exports in terms of imports and is defined as the ratio of export prices to import prices."



For future reference, Wikipedia is an unreliable source. If you are in an educational institution, it is likely you have access to the New Palgrave Dictionary of Economics (the terms of trade article is here). If not, you can google "terms of trade pdf" and hopefully get the slides of a generous professor who is sharing his/her knowledge with the world.